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rl/Lib/site-packages/networkx/algorithms/tree/__init__.py

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+from .branchings import *
+from .coding import *
+from .mst import *
+from .recognition import *
+from .operations import *
+from .decomposition import *

rl/Lib/site-packages/networkx/algorithms/tree/coding.py

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+"""Functions for encoding and decoding trees.
+
+Since a tree is a highly restricted form of graph, it can be represented
+concisely in several ways. This module includes functions for encoding
+and decoding trees in the form of nested tuples and Prüfer
+sequences. The former requires a rooted tree, whereas the latter can be
+applied to unrooted trees. Furthermore, there is a bijection from Prüfer
+sequences to labeled trees.
+
+"""
+
+from collections import Counter
+from itertools import chain
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = [
+    "from_nested_tuple",
+    "from_prufer_sequence",
+    "NotATree",
+    "to_nested_tuple",
+    "to_prufer_sequence",
+]
+
+
+class NotATree(nx.NetworkXException):
+    """Raised when a function expects a tree (that is, a connected
+    undirected graph with no cycles) but gets a non-tree graph as input
+    instead.
+
+    """
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(graphs="T")
+def to_nested_tuple(T, root, canonical_form=False):
+    """Returns a nested tuple representation of the given tree.
+
+    The nested tuple representation of a tree is defined
+    recursively. The tree with one node and no edges is represented by
+    the empty tuple, ``()``. A tree with ``k`` subtrees is represented
+    by a tuple of length ``k`` in which each element is the nested tuple
+    representation of a subtree.
+
+    Parameters
+    ----------
+    T : NetworkX graph
+        An undirected graph object representing a tree.
+
+    root : node
+        The node in ``T`` to interpret as the root of the tree.
+
+    canonical_form : bool
+        If ``True``, each tuple is sorted so that the function returns
+        a canonical form for rooted trees. This means "lighter" subtrees
+        will appear as nested tuples before "heavier" subtrees. In this
+        way, each isomorphic rooted tree has the same nested tuple
+        representation.
+
+    Returns
+    -------
+    tuple
+        A nested tuple representation of the tree.
+
+    Notes
+    -----
+    This function is *not* the inverse of :func:`from_nested_tuple`; the
+    only guarantee is that the rooted trees are isomorphic.
+
+    See also
+    --------
+    from_nested_tuple
+    to_prufer_sequence
+
+    Examples
+    --------
+    The tree need not be a balanced binary tree::
+
+        >>> T = nx.Graph()
+        >>> T.add_edges_from([(0, 1), (0, 2), (0, 3)])
+        >>> T.add_edges_from([(1, 4), (1, 5)])
+        >>> T.add_edges_from([(3, 6), (3, 7)])
+        >>> root = 0
+        >>> nx.to_nested_tuple(T, root)
+        (((), ()), (), ((), ()))
+
+    Continuing the above example, if ``canonical_form`` is ``True``, the
+    nested tuples will be sorted::
+
+        >>> nx.to_nested_tuple(T, root, canonical_form=True)
+        ((), ((), ()), ((), ()))
+
+    Even the path graph can be interpreted as a tree::
+
+        >>> T = nx.path_graph(4)
+        >>> root = 0
+        >>> nx.to_nested_tuple(T, root)
+        ((((),),),)
+
+    """
+
+    def _make_tuple(T, root, _parent):
+        """Recursively compute the nested tuple representation of the
+        given rooted tree.
+
+        ``_parent`` is the parent node of ``root`` in the supertree in
+        which ``T`` is a subtree, or ``None`` if ``root`` is the root of
+        the supertree. This argument is used to determine which
+        neighbors of ``root`` are children and which is the parent.
+
+        """
+        # Get the neighbors of `root` that are not the parent node. We
+        # are guaranteed that `root` is always in `T` by construction.
+        children = set(T[root]) - {_parent}
+        if len(children) == 0:
+            return ()
+        nested = (_make_tuple(T, v, root) for v in children)
+        if canonical_form:
+            nested = sorted(nested)
+        return tuple(nested)
+
+    # Do some sanity checks on the input.
+    if not nx.is_tree(T):
+        raise nx.NotATree("provided graph is not a tree")
+    if root not in T:
+        raise nx.NodeNotFound(f"Graph {T} contains no node {root}")
+
+    return _make_tuple(T, root, None)
+
+
+@nx._dispatchable(graphs=None, returns_graph=True)
+def from_nested_tuple(sequence, sensible_relabeling=False):
+    """Returns the rooted tree corresponding to the given nested tuple.
+
+    The nested tuple representation of a tree is defined
+    recursively. The tree with one node and no edges is represented by
+    the empty tuple, ``()``. A tree with ``k`` subtrees is represented
+    by a tuple of length ``k`` in which each element is the nested tuple
+    representation of a subtree.
+
+    Parameters
+    ----------
+    sequence : tuple
+        A nested tuple representing a rooted tree.
+
+    sensible_relabeling : bool
+        Whether to relabel the nodes of the tree so that nodes are
+        labeled in increasing order according to their breadth-first
+        search order from the root node.
+
+    Returns
+    -------
+    NetworkX graph
+        The tree corresponding to the given nested tuple, whose root
+        node is node 0. If ``sensible_labeling`` is ``True``, nodes will
+        be labeled in breadth-first search order starting from the root
+        node.
+
+    Notes
+    -----
+    This function is *not* the inverse of :func:`to_nested_tuple`; the
+    only guarantee is that the rooted trees are isomorphic.
+
+    See also
+    --------
+    to_nested_tuple
+    from_prufer_sequence
+
+    Examples
+    --------
+    Sensible relabeling ensures that the nodes are labeled from the root
+    starting at 0::
+
+        >>> balanced = (((), ()), ((), ()))
+        >>> T = nx.from_nested_tuple(balanced, sensible_relabeling=True)
+        >>> edges = [(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)]
+        >>> all((u, v) in T.edges() or (v, u) in T.edges() for (u, v) in edges)
+        True
+
+    """
+
+    def _make_tree(sequence):
+        """Recursively creates a tree from the given sequence of nested
+        tuples.
+
+        This function employs the :func:`~networkx.tree.join` function
+        to recursively join subtrees into a larger tree.
+
+        """
+        # The empty sequence represents the empty tree, which is the
+        # (unique) graph with a single node. We mark the single node
+        # with an attribute that indicates that it is the root of the
+        # graph.
+        if len(sequence) == 0:
+            return nx.empty_graph(1)
+        # For a nonempty sequence, get the subtrees for each child
+        # sequence and join all the subtrees at their roots. After
+        # joining the subtrees, the root is node 0.
+        return nx.tree.join_trees([(_make_tree(child), 0) for child in sequence])
+
+    # Make the tree and remove the `is_root` node attribute added by the
+    # helper function.
+    T = _make_tree(sequence)
+    if sensible_relabeling:
+        # Relabel the nodes according to their breadth-first search
+        # order, starting from the root node (that is, the node 0).
+        bfs_nodes = chain([0], (v for u, v in nx.bfs_edges(T, 0)))
+        labels = {v: i for i, v in enumerate(bfs_nodes)}
+        # We would like to use `copy=False`, but `relabel_nodes` doesn't
+        # allow a relabel mapping that can't be topologically sorted.
+        T = nx.relabel_nodes(T, labels)
+    return T
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(graphs="T")
+def to_prufer_sequence(T):
+    r"""Returns the Prüfer sequence of the given tree.
+
+    A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and
+    *n* - 1, inclusive. The tree corresponding to a given Prüfer
+    sequence can be recovered by repeatedly joining a node in the
+    sequence with a node with the smallest potential degree according to
+    the sequence.
+
+    Parameters
+    ----------
+    T : NetworkX graph
+        An undirected graph object representing a tree.
+
+    Returns
+    -------
+    list
+        The Prüfer sequence of the given tree.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If the number of nodes in `T` is less than two.
+
+    NotATree
+        If `T` is not a tree.
+
+    KeyError
+        If the set of nodes in `T` is not {0, …, *n* - 1}.
+
+    Notes
+    -----
+    There is a bijection from labeled trees to Prüfer sequences. This
+    function is the inverse of the :func:`from_prufer_sequence`
+    function.
+
+    Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead
+    of from 0 to *n* - 1. This function requires nodes to be labeled in
+    the latter form. You can use :func:`~networkx.relabel_nodes` to
+    relabel the nodes of your tree to the appropriate format.
+
+    This implementation is from [1]_ and has a running time of
+    $O(n)$.
+
+    See also
+    --------
+    to_nested_tuple
+    from_prufer_sequence
+
+    References
+    ----------
+    .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.
+           "An optimal algorithm for Prufer codes."
+           *Journal of Software Engineering and Applications* 2.02 (2009): 111.
+           <https://doi.org/10.4236/jsea.2009.22016>
+
+    Examples
+    --------
+    There is a bijection between Prüfer sequences and labeled trees, so
+    this function is the inverse of the :func:`from_prufer_sequence`
+    function:
+
+    >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
+    >>> tree = nx.Graph(edges)
+    >>> sequence = nx.to_prufer_sequence(tree)
+    >>> sequence
+    [3, 3, 3, 4]
+    >>> tree2 = nx.from_prufer_sequence(sequence)
+    >>> list(tree2.edges()) == edges
+    True
+
+    """
+    # Perform some sanity checks on the input.
+    n = len(T)
+    if n < 2:
+        msg = "Prüfer sequence undefined for trees with fewer than two nodes"
+        raise nx.NetworkXPointlessConcept(msg)
+    if not nx.is_tree(T):
+        raise nx.NotATree("provided graph is not a tree")
+    if set(T) != set(range(n)):
+        raise KeyError("tree must have node labels {0, ..., n - 1}")
+
+    degree = dict(T.degree())
+
+    def parents(u):
+        return next(v for v in T[u] if degree[v] > 1)
+
+    index = u = next(k for k in range(n) if degree[k] == 1)
+    result = []
+    for i in range(n - 2):
+        v = parents(u)
+        result.append(v)
+        degree[v] -= 1
+        if v < index and degree[v] == 1:
+            u = v
+        else:
+            index = u = next(k for k in range(index + 1, n) if degree[k] == 1)
+    return result
+
+
+@nx._dispatchable(graphs=None, returns_graph=True)
+def from_prufer_sequence(sequence):
+    r"""Returns the tree corresponding to the given Prüfer sequence.
+
+    A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and
+    *n* - 1, inclusive. The tree corresponding to a given Prüfer
+    sequence can be recovered by repeatedly joining a node in the
+    sequence with a node with the smallest potential degree according to
+    the sequence.
+
+    Parameters
+    ----------
+    sequence : list
+        A Prüfer sequence, which is a list of *n* - 2 integers between
+        zero and *n* - 1, inclusive.
+
+    Returns
+    -------
+    NetworkX graph
+        The tree corresponding to the given Prüfer sequence.
+
+    Raises
+    ------
+    NetworkXError
+        If the Prüfer sequence is not valid.
+
+    Notes
+    -----
+    There is a bijection from labeled trees to Prüfer sequences. This
+    function is the inverse of the :func:`from_prufer_sequence` function.
+
+    Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead
+    of from 0 to *n* - 1. This function requires nodes to be labeled in
+    the latter form. You can use :func:`networkx.relabel_nodes` to
+    relabel the nodes of your tree to the appropriate format.
+
+    This implementation is from [1]_ and has a running time of
+    $O(n)$.
+
+    References
+    ----------
+    .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.
+           "An optimal algorithm for Prufer codes."
+           *Journal of Software Engineering and Applications* 2.02 (2009): 111.
+           <https://doi.org/10.4236/jsea.2009.22016>
+
+    See also
+    --------
+    from_nested_tuple
+    to_prufer_sequence
+
+    Examples
+    --------
+    There is a bijection between Prüfer sequences and labeled trees, so
+    this function is the inverse of the :func:`to_prufer_sequence`
+    function:
+
+    >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
+    >>> tree = nx.Graph(edges)
+    >>> sequence = nx.to_prufer_sequence(tree)
+    >>> sequence
+    [3, 3, 3, 4]
+    >>> tree2 = nx.from_prufer_sequence(sequence)
+    >>> list(tree2.edges()) == edges
+    True
+
+    """
+    n = len(sequence) + 2
+    # `degree` stores the remaining degree (plus one) for each node. The
+    # degree of a node in the decoded tree is one more than the number
+    # of times it appears in the code.
+    degree = Counter(chain(sequence, range(n)))
+    T = nx.empty_graph(n)
+    # `not_orphaned` is the set of nodes that have a parent in the
+    # tree. After the loop, there should be exactly two nodes that are
+    # not in this set.
+    not_orphaned = set()
+    index = u = next(k for k in range(n) if degree[k] == 1)
+    for v in sequence:
+        # check the validity of the prufer sequence
+        if v < 0 or v > n - 1:
+            raise nx.NetworkXError(
+                f"Invalid Prufer sequence: Values must be between 0 and {n-1}, got {v}"
+            )
+        T.add_edge(u, v)
+        not_orphaned.add(u)
+        degree[v] -= 1
+        if v < index and degree[v] == 1:
+            u = v
+        else:
+            index = u = next(k for k in range(index + 1, n) if degree[k] == 1)
+    # At this point, there must be exactly two orphaned nodes; join them.
+    orphans = set(T) - not_orphaned
+    u, v = orphans
+    T.add_edge(u, v)
+    return T

rl/Lib/site-packages/networkx/algorithms/tree/decomposition.py

@@ -0,0 +1,88 @@
+r"""Function for computing a junction tree of a graph."""
+
+from itertools import combinations
+
+import networkx as nx
+from networkx.algorithms import chordal_graph_cliques, complete_to_chordal_graph, moral
+from networkx.utils import not_implemented_for
+
+__all__ = ["junction_tree"]
+
+
+@not_implemented_for("multigraph")
+@nx._dispatchable(returns_graph=True)
+def junction_tree(G):
+    r"""Returns a junction tree of a given graph.
+
+    A junction tree (or clique tree) is constructed from a (un)directed graph G.
+    The tree is constructed based on a moralized and triangulated version of G.
+    The tree's nodes consist of maximal cliques and sepsets of the revised graph.
+    The sepset of two cliques is the intersection of the nodes of these cliques,
+    e.g. the sepset of (A,B,C) and (A,C,E,F) is (A,C). These nodes are often called
+    "variables" in this literature. The tree is bipartite with each sepset
+    connected to its two cliques.
+
+    Junction Trees are not unique as the order of clique consideration determines
+    which sepsets are included.
+
+    The junction tree algorithm consists of five steps [1]_:
+
+    1. Moralize the graph
+    2. Triangulate the graph
+    3. Find maximal cliques
+    4. Build the tree from cliques, connecting cliques with shared
+       nodes, set edge-weight to number of shared variables
+    5. Find maximum spanning tree
+
+
+    Parameters
+    ----------
+    G : networkx.Graph
+        Directed or undirected graph.
+
+    Returns
+    -------
+    junction_tree : networkx.Graph
+        The corresponding junction tree of `G`.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        Raised if `G` is an instance of `MultiGraph` or `MultiDiGraph`.
+
+    References
+    ----------
+    .. [1] Junction tree algorithm:
+       https://en.wikipedia.org/wiki/Junction_tree_algorithm
+
+    .. [2] Finn V. Jensen and Frank Jensen. 1994. Optimal
+       junction trees. In Proceedings of the Tenth international
+       conference on Uncertainty in artificial intelligence (UAI’94).
+       Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 360–366.
+    """
+
+    clique_graph = nx.Graph()
+
+    if G.is_directed():
+        G = moral.moral_graph(G)
+    chordal_graph, _ = complete_to_chordal_graph(G)
+
+    cliques = [tuple(sorted(i)) for i in chordal_graph_cliques(chordal_graph)]
+    clique_graph.add_nodes_from(cliques, type="clique")
+
+    for edge in combinations(cliques, 2):
+        set_edge_0 = set(edge[0])
+        set_edge_1 = set(edge[1])
+        if not set_edge_0.isdisjoint(set_edge_1):
+            sepset = tuple(sorted(set_edge_0.intersection(set_edge_1)))
+            clique_graph.add_edge(edge[0], edge[1], weight=len(sepset), sepset=sepset)
+
+    junction_tree = nx.maximum_spanning_tree(clique_graph)
+
+    for edge in list(junction_tree.edges(data=True)):
+        junction_tree.add_node(edge[2]["sepset"], type="sepset")
+        junction_tree.add_edge(edge[0], edge[2]["sepset"])
+        junction_tree.add_edge(edge[1], edge[2]["sepset"])
+        junction_tree.remove_edge(edge[0], edge[1])
+
+    return junction_tree

rl/Lib/site-packages/networkx/algorithms/tree/operations.py

@@ -0,0 +1,105 @@
+"""Operations on trees."""
+
+from functools import partial
+from itertools import accumulate, chain
+
+import networkx as nx
+
+__all__ = ["join_trees"]
+
+
+# Argument types don't match dispatching, but allow manual selection of backend
+@nx._dispatchable(graphs=None, returns_graph=True)
+def join_trees(rooted_trees, *, label_attribute=None, first_label=0):
+    """Returns a new rooted tree made by joining `rooted_trees`
+
+    Constructs a new tree by joining each tree in `rooted_trees`.
+    A new root node is added and connected to each of the roots
+    of the input trees. While copying the nodes from the trees,
+    relabeling to integers occurs. If the `label_attribute` is provided,
+    the old node labels will be stored in the new tree under this attribute.
+
+    Parameters
+    ----------
+    rooted_trees : list
+        A list of pairs in which each left element is a NetworkX graph
+        object representing a tree and each right element is the root
+        node of that tree. The nodes of these trees will be relabeled to
+        integers.
+
+    label_attribute : str
+        If provided, the old node labels will be stored in the new tree
+        under this node attribute. If not provided, the original labels
+        of the nodes in the input trees are not stored.
+
+    first_label : int, optional (default=0)
+        Specifies the label for the new root node. If provided, the root node of the joined tree
+        will have this label. If not provided, the root node will default to a label of 0.
+
+    Returns
+    -------
+    NetworkX graph
+        The rooted tree resulting from joining the provided `rooted_trees`. The new tree has a root node
+        labeled as specified by `first_label` (defaulting to 0 if not provided). Subtrees from the input
+        `rooted_trees` are attached to this new root node. Each non-root node, if the `label_attribute`
+        is provided, has an attribute that indicates the original label of the node in the input tree.
+
+    Notes
+    -----
+    Trees are stored in NetworkX as NetworkX Graphs. There is no specific
+    enforcement of the fact that these are trees. Testing for each tree
+    can be done using :func:`networkx.is_tree`.
+
+    Graph, edge, and node attributes are propagated from the given
+    rooted trees to the created tree. If there are any overlapping graph
+    attributes, those from later trees will overwrite those from earlier
+    trees in the tuple of positional arguments.
+
+    Examples
+    --------
+    Join two full balanced binary trees of height *h* to get a full
+    balanced binary tree of depth *h* + 1::
+
+        >>> h = 4
+        >>> left = nx.balanced_tree(2, h)
+        >>> right = nx.balanced_tree(2, h)
+        >>> joined_tree = nx.join_trees([(left, 0), (right, 0)])
+        >>> nx.is_isomorphic(joined_tree, nx.balanced_tree(2, h + 1))
+        True
+
+    """
+    if not rooted_trees:
+        return nx.empty_graph(1)
+
+    # Unzip the zipped list of (tree, root) pairs.
+    trees, roots = zip(*rooted_trees)
+
+    # The join of the trees has the same type as the type of the first tree.
+    R = type(trees[0])()
+
+    lengths = (len(tree) for tree in trees[:-1])
+    first_labels = list(accumulate(lengths, initial=first_label + 1))
+
+    new_roots = []
+    for tree, root, first_node in zip(trees, roots, first_labels):
+        new_root = first_node + list(tree.nodes()).index(root)
+        new_roots.append(new_root)
+
+    # Relabel the nodes so that their union is the integers starting at first_label.
+    relabel = partial(
+        nx.convert_node_labels_to_integers, label_attribute=label_attribute
+    )
+    new_trees = [
+        relabel(tree, first_label=first_label)
+        for tree, first_label in zip(trees, first_labels)
+    ]
+
+    # Add all sets of nodes and edges, attributes
+    for tree in new_trees:
+        R.update(tree)
+
+    # Finally, join the subtrees at the root. We know first_label is unused by the way we relabeled the subtrees.
+    R.add_node(first_label)
+    R.add_edges_from((first_label, root) for root in new_roots)
+
+    return R

rl/Lib/site-packages/networkx/algorithms/tree/recognition.py

@@ -0,0 +1,273 @@
+"""
+Recognition Tests
+=================
+
+A *forest* is an acyclic, undirected graph, and a *tree* is a connected forest.
+Depending on the subfield, there are various conventions for generalizing these
+definitions to directed graphs.
+
+In one convention, directed variants of forest and tree are defined in an
+identical manner, except that the direction of the edges is ignored. In effect,
+each directed edge is treated as a single undirected edge. Then, additional
+restrictions are imposed to define *branchings* and *arborescences*.
+
+In another convention, directed variants of forest and tree correspond to
+the previous convention's branchings and arborescences, respectively. Then two
+new terms, *polyforest* and *polytree*, are defined to correspond to the other
+convention's forest and tree.
+
+Summarizing::
+
+   +-----------------------------+
+   | Convention A | Convention B |
+   +=============================+
+   | forest       | polyforest   |
+   | tree         | polytree     |
+   | branching    | forest       |
+   | arborescence | tree         |
+   +-----------------------------+
+
+Each convention has its reasons. The first convention emphasizes definitional
+similarity in that directed forests and trees are only concerned with
+acyclicity and do not have an in-degree constraint, just as their undirected
+counterparts do not. The second convention emphasizes functional similarity
+in the sense that the directed analog of a spanning tree is a spanning
+arborescence. That is, take any spanning tree and choose one node as the root.
+Then every edge is assigned a direction such there is a directed path from the
+root to every other node. The result is a spanning arborescence.
+
+NetworkX follows convention "A". Explicitly, these are:
+
+undirected forest
+   An undirected graph with no undirected cycles.
+
+undirected tree
+   A connected, undirected forest.
+
+directed forest
+   A directed graph with no undirected cycles. Equivalently, the underlying
+   graph structure (which ignores edge orientations) is an undirected forest.
+   In convention B, this is known as a polyforest.
+
+directed tree
+   A weakly connected, directed forest. Equivalently, the underlying graph
+   structure (which ignores edge orientations) is an undirected tree. In
+   convention B, this is known as a polytree.
+
+branching
+   A directed forest with each node having, at most, one parent. So the maximum
+   in-degree is equal to 1. In convention B, this is known as a forest.
+
+arborescence
+   A directed tree with each node having, at most, one parent. So the maximum
+   in-degree is equal to 1. In convention B, this is known as a tree.
+
+For trees and arborescences, the adjective "spanning" may be added to designate
+that the graph, when considered as a forest/branching, consists of a single
+tree/arborescence that includes all nodes in the graph. It is true, by
+definition, that every tree/arborescence is spanning with respect to the nodes
+that define the tree/arborescence and so, it might seem redundant to introduce
+the notion of "spanning". However, the nodes may represent a subset of
+nodes from a larger graph, and it is in this context that the term "spanning"
+becomes a useful notion.
+
+"""
+
+import networkx as nx
+
+__all__ = ["is_arborescence", "is_branching", "is_forest", "is_tree"]
+
+
+@nx.utils.not_implemented_for("undirected")
+@nx._dispatchable
+def is_arborescence(G):
+    """
+    Returns True if `G` is an arborescence.
+
+    An arborescence is a directed tree with maximum in-degree equal to 1.
+
+    Parameters
+    ----------
+    G : graph
+        The graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is an arborescence.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (2, 3), (3, 4)])
+    >>> nx.is_arborescence(G)
+    True
+    >>> G.remove_edge(0, 1)
+    >>> G.add_edge(1, 2)  # maximum in-degree is 2
+    >>> nx.is_arborescence(G)
+    False
+
+    Notes
+    -----
+    In another convention, an arborescence is known as a *tree*.
+
+    See Also
+    --------
+    is_tree
+
+    """
+    return is_tree(G) and max(d for n, d in G.in_degree()) <= 1
+
+
+@nx.utils.not_implemented_for("undirected")
+@nx._dispatchable
+def is_branching(G):
+    """
+    Returns True if `G` is a branching.
+
+    A branching is a directed forest with maximum in-degree equal to 1.
+
+    Parameters
+    ----------
+    G : directed graph
+        The directed graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is a branching.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 3), (3, 4)])
+    >>> nx.is_branching(G)
+    True
+    >>> G.remove_edge(2, 3)
+    >>> G.add_edge(3, 1)  # maximum in-degree is 2
+    >>> nx.is_branching(G)
+    False
+
+    Notes
+    -----
+    In another convention, a branching is also known as a *forest*.
+
+    See Also
+    --------
+    is_forest
+
+    """
+    return is_forest(G) and max(d for n, d in G.in_degree()) <= 1
+
+
+@nx._dispatchable
+def is_forest(G):
+    """
+    Returns True if `G` is a forest.
+
+    A forest is a graph with no undirected cycles.
+
+    For directed graphs, `G` is a forest if the underlying graph is a forest.
+    The underlying graph is obtained by treating each directed edge as a single
+    undirected edge in a multigraph.
+
+    Parameters
+    ----------
+    G : graph
+        The graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is a forest.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `G` is empty.
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_edges_from([(1, 2), (1, 3), (2, 4), (2, 5)])
+    >>> nx.is_forest(G)
+    True
+    >>> G.add_edge(4, 1)
+    >>> nx.is_forest(G)
+    False
+
+    Notes
+    -----
+    In another convention, a directed forest is known as a *polyforest* and
+    then *forest* corresponds to a *branching*.
+
+    See Also
+    --------
+    is_branching
+
+    """
+    if len(G) == 0:
+        raise nx.exception.NetworkXPointlessConcept("G has no nodes.")
+
+    if G.is_directed():
+        components = (G.subgraph(c) for c in nx.weakly_connected_components(G))
+    else:
+        components = (G.subgraph(c) for c in nx.connected_components(G))
+
+    return all(len(c) - 1 == c.number_of_edges() for c in components)
+
+
+@nx._dispatchable
+def is_tree(G):
+    """
+    Returns True if `G` is a tree.
+
+    A tree is a connected graph with no undirected cycles.
+
+    For directed graphs, `G` is a tree if the underlying graph is a tree. The
+    underlying graph is obtained by treating each directed edge as a single
+    undirected edge in a multigraph.
+
+    Parameters
+    ----------
+    G : graph
+        The graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is a tree.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `G` is empty.
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_edges_from([(1, 2), (1, 3), (2, 4), (2, 5)])
+    >>> nx.is_tree(G)  # n-1 edges
+    True
+    >>> G.add_edge(3, 4)
+    >>> nx.is_tree(G)  # n edges
+    False
+
+    Notes
+    -----
+    In another convention, a directed tree is known as a *polytree* and then
+    *tree* corresponds to an *arborescence*.
+
+    See Also
+    --------
+    is_arborescence
+
+    """
+    if len(G) == 0:
+        raise nx.exception.NetworkXPointlessConcept("G has no nodes.")
+
+    if G.is_directed():
+        is_connected = nx.is_weakly_connected
+    else:
+        is_connected = nx.is_connected
+
+    # A connected graph with no cycles has n-1 edges.
+    return len(G) - 1 == G.number_of_edges() and is_connected(G)

rl/Lib/site-packages/networkx/algorithms/tree/tests/test_branchings.py

@@ -0,0 +1,624 @@
+import math
+from operator import itemgetter
+
+import pytest
+
+np = pytest.importorskip("numpy")
+
+import networkx as nx
+from networkx.algorithms.tree import branchings, recognition
+
+#
+# Explicitly discussed examples from Edmonds paper.
+#
+
+# Used in Figures A-F.
+#
+# fmt: off
+G_array = np.array([
+    # 0   1   2   3   4   5   6   7   8
+    [0, 0, 12, 0, 12, 0, 0, 0, 0],  # 0
+    [4, 0, 0, 0, 0, 13, 0, 0, 0],  # 1
+    [0, 17, 0, 21, 0, 12, 0, 0, 0],  # 2
+    [5, 0, 0, 0, 17, 0, 18, 0, 0],  # 3
+    [0, 0, 0, 0, 0, 0, 0, 12, 0],  # 4
+    [0, 0, 0, 0, 0, 0, 14, 0, 12],  # 5
+    [0, 0, 21, 0, 0, 0, 0, 0, 15],  # 6
+    [0, 0, 0, 19, 0, 0, 15, 0, 0],  # 7
+    [0, 0, 0, 0, 0, 0, 0, 18, 0],  # 8
+], dtype=int)
+
+# Two copies of the graph from the original paper as disconnected components
+G_big_array = np.zeros(np.array(G_array.shape) * 2, dtype=int)
+G_big_array[:G_array.shape[0], :G_array.shape[1]] = G_array
+G_big_array[G_array.shape[0]:, G_array.shape[1]:] = G_array
+
+# fmt: on
+
+
+def G1():
+    G = nx.from_numpy_array(G_array, create_using=nx.MultiDiGraph)
+    return G
+
+
+def G2():
+    # Now we shift all the weights by -10.
+    # Should not affect optimal arborescence, but does affect optimal branching.
+    Garr = G_array.copy()
+    Garr[np.nonzero(Garr)] -= 10
+    G = nx.from_numpy_array(Garr, create_using=nx.MultiDiGraph)
+    return G
+
+
+# An optimal branching for G1 that is also a spanning arborescence. So it is
+# also an optimal spanning arborescence.
+#
+optimal_arborescence_1 = [
+    (0, 2, 12),
+    (2, 1, 17),
+    (2, 3, 21),
+    (1, 5, 13),
+    (3, 4, 17),
+    (3, 6, 18),
+    (6, 8, 15),
+    (8, 7, 18),
+]
+
+# For G2, the optimal branching of G1 (with shifted weights) is no longer
+# an optimal branching, but it is still an optimal spanning arborescence
+# (just with shifted weights). An optimal branching for G2 is similar to what
+# appears in figure G (this is greedy_subopt_branching_1a below), but with the
+# edge (3, 0, 5), which is now (3, 0, -5), removed. Thus, the optimal branching
+# is not a spanning arborescence. The code finds optimal_branching_2a.
+# An alternative and equivalent branching is optimal_branching_2b. We would
+# need to modify the code to iterate through all equivalent optimal branchings.
+#
+# These are maximal branchings or arborescences.
+optimal_branching_2a = [
+    (5, 6, 4),
+    (6, 2, 11),
+    (6, 8, 5),
+    (8, 7, 8),
+    (2, 1, 7),
+    (2, 3, 11),
+    (3, 4, 7),
+]
+optimal_branching_2b = [
+    (8, 7, 8),
+    (7, 3, 9),
+    (3, 4, 7),
+    (3, 6, 8),
+    (6, 2, 11),
+    (2, 1, 7),
+    (1, 5, 3),
+]
+optimal_arborescence_2 = [
+    (0, 2, 2),
+    (2, 1, 7),
+    (2, 3, 11),
+    (1, 5, 3),
+    (3, 4, 7),
+    (3, 6, 8),
+    (6, 8, 5),
+    (8, 7, 8),
+]
+
+# Two suboptimal maximal branchings on G1 obtained from a greedy algorithm.
+# 1a matches what is shown in Figure G in Edmonds's paper.
+greedy_subopt_branching_1a = [
+    (5, 6, 14),
+    (6, 2, 21),
+    (6, 8, 15),
+    (8, 7, 18),
+    (2, 1, 17),
+    (2, 3, 21),
+    (3, 0, 5),
+    (3, 4, 17),
+]
+greedy_subopt_branching_1b = [
+    (8, 7, 18),
+    (7, 6, 15),
+    (6, 2, 21),
+    (2, 1, 17),
+    (2, 3, 21),
+    (1, 5, 13),
+    (3, 0, 5),
+    (3, 4, 17),
+]
+
+
+def build_branching(edges, double=False):
+    G = nx.DiGraph()
+    for u, v, weight in edges:
+        G.add_edge(u, v, weight=weight)
+        if double:
+            G.add_edge(u + 9, v + 9, weight=weight)
+    return G
+
+
+def sorted_edges(G, attr="weight", default=1):
+    edges = [(u, v, data.get(attr, default)) for (u, v, data) in G.edges(data=True)]
+    edges = sorted(edges, key=lambda x: (x[2], x[1], x[0]))
+    return edges
+
+
+def assert_equal_branchings(G1, G2, attr="weight", default=1):
+    edges1 = list(G1.edges(data=True))
+    edges2 = list(G2.edges(data=True))
+    assert len(edges1) == len(edges2)
+
+    # Grab the weights only.
+    e1 = sorted_edges(G1, attr, default)
+    e2 = sorted_edges(G2, attr, default)
+
+    for a, b in zip(e1, e2):
+        assert a[:2] == b[:2]
+        np.testing.assert_almost_equal(a[2], b[2])
+
+
+################
+
+
+def test_optimal_branching1():
+    G = build_branching(optimal_arborescence_1)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 131
+
+
+def test_optimal_branching2a():
+    G = build_branching(optimal_branching_2a)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 53
+
+
+def test_optimal_branching2b():
+    G = build_branching(optimal_branching_2b)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 53
+
+
+def test_optimal_arborescence2():
+    G = build_branching(optimal_arborescence_2)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 51
+
+
+def test_greedy_suboptimal_branching1a():
+    G = build_branching(greedy_subopt_branching_1a)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 128
+
+
+def test_greedy_suboptimal_branching1b():
+    G = build_branching(greedy_subopt_branching_1b)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 127
+
+
+def test_greedy_max1():
+    # Standard test.
+    #
+    G = G1()
+    B = branchings.greedy_branching(G)
+    # There are only two possible greedy branchings. The sorting is such
+    # that it should equal the second suboptimal branching: 1b.
+    B_ = build_branching(greedy_subopt_branching_1b)
+    assert_equal_branchings(B, B_)
+
+
+def test_greedy_branching_kwarg_kind():
+    G = G1()
+    with pytest.raises(nx.NetworkXException, match="Unknown value for `kind`."):
+        B = branchings.greedy_branching(G, kind="lol")
+
+
+def test_greedy_branching_for_unsortable_nodes():
+    G = nx.DiGraph()
+    G.add_weighted_edges_from([((2, 3), 5, 1), (3, "a", 1), (2, 4, 5)])
+    edges = [(u, v, data.get("weight", 1)) for (u, v, data) in G.edges(data=True)]
+    with pytest.raises(TypeError):
+        edges.sort(key=itemgetter(2, 0, 1), reverse=True)
+    B = branchings.greedy_branching(G, kind="max").edges(data=True)
+    assert list(B) == [
+        ((2, 3), 5, {"weight": 1}),
+        (3, "a", {"weight": 1}),
+        (2, 4, {"weight": 5}),
+    ]
+
+
+def test_greedy_max2():
+    # Different default weight.
+    #
+    G = G1()
+    del G[1][0][0]["weight"]
+    B = branchings.greedy_branching(G, default=6)
+    # Chosen so that edge (3,0,5) is not selected and (1,0,6) is instead.
+
+    edges = [
+        (1, 0, 6),
+        (1, 5, 13),
+        (7, 6, 15),
+        (2, 1, 17),
+        (3, 4, 17),
+        (8, 7, 18),
+        (2, 3, 21),
+        (6, 2, 21),
+    ]
+    B_ = build_branching(edges)
+    assert_equal_branchings(B, B_)
+
+
+def test_greedy_max3():
+    # All equal weights.
+    #
+    G = G1()
+    B = branchings.greedy_branching(G, attr=None)
+
+    # This is mostly arbitrary...the output was generated by running the algo.
+    edges = [
+        (2, 1, 1),
+        (3, 0, 1),
+        (3, 4, 1),
+        (5, 8, 1),
+        (6, 2, 1),
+        (7, 3, 1),
+        (7, 6, 1),
+        (8, 7, 1),
+    ]
+    B_ = build_branching(edges)
+    assert_equal_branchings(B, B_, default=1)
+
+
+def test_greedy_min():
+    G = G1()
+    B = branchings.greedy_branching(G, kind="min")
+
+    edges = [
+        (1, 0, 4),
+        (0, 2, 12),
+        (0, 4, 12),
+        (2, 5, 12),
+        (4, 7, 12),
+        (5, 8, 12),
+        (5, 6, 14),
+        (7, 3, 19),
+    ]
+    B_ = build_branching(edges)
+    assert_equal_branchings(B, B_)
+
+
+def test_edmonds1_maxbranch():
+    G = G1()
+    x = branchings.maximum_branching(G)
+    x_ = build_branching(optimal_arborescence_1)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds1_maxarbor():
+    G = G1()
+    x = branchings.maximum_spanning_arborescence(G)
+    x_ = build_branching(optimal_arborescence_1)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds1_minimal_branching():
+    # graph will have something like a minimum arborescence but no spanning one
+    G = nx.from_numpy_array(G_big_array, create_using=nx.DiGraph)
+    B = branchings.minimal_branching(G)
+    edges = [
+        (3, 0, 5),
+        (0, 2, 12),
+        (0, 4, 12),
+        (2, 5, 12),
+        (4, 7, 12),
+        (5, 8, 12),
+        (5, 6, 14),
+        (2, 1, 17),
+    ]
+    B_ = build_branching(edges, double=True)
+    assert_equal_branchings(B, B_)
+
+
+def test_edmonds2_maxbranch():
+    G = G2()
+    x = branchings.maximum_branching(G)
+    x_ = build_branching(optimal_branching_2a)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds2_maxarbor():
+    G = G2()
+    x = branchings.maximum_spanning_arborescence(G)
+    x_ = build_branching(optimal_arborescence_2)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds2_minarbor():
+    G = G1()
+    x = branchings.minimum_spanning_arborescence(G)
+    # This was obtained from algorithm. Need to verify it independently.
+    # Branch weight is: 96
+    edges = [
+        (3, 0, 5),
+        (0, 2, 12),
+        (0, 4, 12),
+        (2, 5, 12),
+        (4, 7, 12),
+        (5, 8, 12),
+        (5, 6, 14),
+        (2, 1, 17),
+    ]
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds3_minbranch1():
+    G = G1()
+    x = branchings.minimum_branching(G)
+    edges = []
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds3_minbranch2():
+    G = G1()
+    G.add_edge(8, 9, weight=-10)
+    x = branchings.minimum_branching(G)
+    edges = [(8, 9, -10)]
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+# Need more tests
+
+
+def test_mst():
+    # Make sure we get the same results for undirected graphs.
+    # Example from: https://en.wikipedia.org/wiki/Kruskal's_algorithm
+    G = nx.Graph()
+    edgelist = [
+        (0, 3, [("weight", 5)]),
+        (0, 1, [("weight", 7)]),
+        (1, 3, [("weight", 9)]),
+        (1, 2, [("weight", 8)]),
+        (1, 4, [("weight", 7)]),
+        (3, 4, [("weight", 15)]),
+        (3, 5, [("weight", 6)]),
+        (2, 4, [("weight", 5)]),
+        (4, 5, [("weight", 8)]),
+        (4, 6, [("weight", 9)]),
+        (5, 6, [("weight", 11)]),
+    ]
+    G.add_edges_from(edgelist)
+    G = G.to_directed()
+    x = branchings.minimum_spanning_arborescence(G)
+
+    edges = [
+        ({0, 1}, 7),
+        ({0, 3}, 5),
+        ({3, 5}, 6),
+        ({1, 4}, 7),
+        ({4, 2}, 5),
+        ({4, 6}, 9),
+    ]
+
+    assert x.number_of_edges() == len(edges)
+    for u, v, d in x.edges(data=True):
+        assert ({u, v}, d["weight"]) in edges
+
+
+def test_mixed_nodetypes():
+    # Smoke test to make sure no TypeError is raised for mixed node types.
+    G = nx.Graph()
+    edgelist = [(0, 3, [("weight", 5)]), (0, "1", [("weight", 5)])]
+    G.add_edges_from(edgelist)
+    G = G.to_directed()
+    x = branchings.minimum_spanning_arborescence(G)
+
+
+def test_edmonds1_minbranch():
+    # Using -G_array and min should give the same as optimal_arborescence_1,
+    # but with all edges negative.
+    edges = [(u, v, -w) for (u, v, w) in optimal_arborescence_1]
+
+    G = nx.from_numpy_array(-G_array, create_using=nx.DiGraph)
+
+    # Quickly make sure max branching is empty.
+    x = branchings.maximum_branching(G)
+    x_ = build_branching([])
+    assert_equal_branchings(x, x_)
+
+    # Now test the min branching.
+    x = branchings.minimum_branching(G)
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+def test_edge_attribute_preservation_normal_graph():
+    # Test that edge attributes are preserved when finding an optimum graph
+    # using the Edmonds class for normal graphs.
+    G = nx.Graph()
+
+    edgelist = [
+        (0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
+        (0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
+        (1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
+    ]
+    G.add_edges_from(edgelist)
+
+    B = branchings.maximum_branching(G, preserve_attrs=True)
+
+    assert B[0][1]["otherattr"] == 1
+    assert B[0][1]["otherattr2"] == 3
+
+
+def test_edge_attribute_preservation_multigraph():
+    # Test that edge attributes are preserved when finding an optimum graph
+    # using the Edmonds class for multigraphs.
+    G = nx.MultiGraph()
+
+    edgelist = [
+        (0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
+        (0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
+        (1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
+    ]
+    G.add_edges_from(edgelist * 2)  # Make sure we have duplicate edge paths
+
+    B = branchings.maximum_branching(G, preserve_attrs=True)
+
+    assert B[0][1][0]["otherattr"] == 1
+    assert B[0][1][0]["otherattr2"] == 3
+
+
+def test_edge_attribute_discard():
+    # Test that edge attributes are discarded if we do not specify to keep them
+    G = nx.Graph()
+
+    edgelist = [
+        (0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
+        (0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
+        (1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
+    ]
+    G.add_edges_from(edgelist)
+
+    B = branchings.maximum_branching(G, preserve_attrs=False)
+
+    edge_dict = B[0][1]
+    with pytest.raises(KeyError):
+        _ = edge_dict["otherattr"]
+
+
+def test_partition_spanning_arborescence():
+    """
+    Test that we can generate minimum spanning arborescences which respect the
+    given partition.
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+    G[3][0]["partition"] = nx.EdgePartition.EXCLUDED
+    G[2][3]["partition"] = nx.EdgePartition.INCLUDED
+    G[7][3]["partition"] = nx.EdgePartition.EXCLUDED
+    G[0][2]["partition"] = nx.EdgePartition.EXCLUDED
+    G[6][2]["partition"] = nx.EdgePartition.INCLUDED
+
+    actual_edges = [
+        (0, 4, 12),
+        (1, 0, 4),
+        (1, 5, 13),
+        (2, 3, 21),
+        (4, 7, 12),
+        (5, 6, 14),
+        (5, 8, 12),
+        (6, 2, 21),
+    ]
+
+    B = branchings.minimum_spanning_arborescence(G, partition="partition")
+    assert_equal_branchings(build_branching(actual_edges), B)
+
+
+def test_arborescence_iterator_min():
+    """
+    Tests the arborescence iterator.
+
+    A brute force method found 680 arborescences in this graph.
+    This test will not verify all of them individually, but will check two
+    things
+
+    * The iterator returns 680 arborescences
+    * The weight of the arborescences is non-strictly increasing
+
+    for more information please visit
+    https://mjschwenne.github.io/2021/06/10/implementing-the-iterators.html
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+
+    arborescence_count = 0
+    arborescence_weight = -math.inf
+    for B in branchings.ArborescenceIterator(G):
+        arborescence_count += 1
+        new_arborescence_weight = B.size(weight="weight")
+        assert new_arborescence_weight >= arborescence_weight
+        arborescence_weight = new_arborescence_weight
+
+    assert arborescence_count == 680
+
+
+def test_arborescence_iterator_max():
+    """
+    Tests the arborescence iterator.
+
+    A brute force method found 680 arborescences in this graph.
+    This test will not verify all of them individually, but will check two
+    things
+
+    * The iterator returns 680 arborescences
+    * The weight of the arborescences is non-strictly decreasing
+
+    for more information please visit
+    https://mjschwenne.github.io/2021/06/10/implementing-the-iterators.html
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+
+    arborescence_count = 0
+    arborescence_weight = math.inf
+    for B in branchings.ArborescenceIterator(G, minimum=False):
+        arborescence_count += 1
+        new_arborescence_weight = B.size(weight="weight")
+        assert new_arborescence_weight <= arborescence_weight
+        arborescence_weight = new_arborescence_weight
+
+    assert arborescence_count == 680
+
+
+def test_arborescence_iterator_initial_partition():
+    """
+    Tests the arborescence iterator with three included edges and three excluded
+    in the initial partition.
+
+    A brute force method similar to the one used in the above tests found that
+    there are 16 arborescences which contain the included edges and not the
+    excluded edges.
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+    included_edges = [(1, 0), (5, 6), (8, 7)]
+    excluded_edges = [(0, 2), (3, 6), (1, 5)]
+
+    arborescence_count = 0
+    arborescence_weight = -math.inf
+    for B in branchings.ArborescenceIterator(
+        G, init_partition=(included_edges, excluded_edges)
+    ):
+        arborescence_count += 1
+        new_arborescence_weight = B.size(weight="weight")
+        assert new_arborescence_weight >= arborescence_weight
+        arborescence_weight = new_arborescence_weight
+        for e in included_edges:
+            assert e in B.edges
+        for e in excluded_edges:
+            assert e not in B.edges
+    assert arborescence_count == 16
+
+
+def test_branchings_with_default_weights():
+    """
+    Tests that various branching algorithms work on graphs without weights.
+    For more information, see issue #7279.
+    """
+    graph = nx.erdos_renyi_graph(10, p=0.2, directed=True, seed=123)
+
+    assert all(
+        "weight" not in d for (u, v, d) in graph.edges(data=True)
+    ), "test is for graphs without a weight attribute"
+
+    # Calling these functions will modify graph inplace to add weights
+    # copy the graph to avoid this.
+    nx.minimum_spanning_arborescence(graph.copy())
+    nx.maximum_spanning_arborescence(graph.copy())
+    nx.minimum_branching(graph.copy())
+    nx.maximum_branching(graph.copy())
+    nx.algorithms.tree.minimal_branching(graph.copy())
+    nx.algorithms.tree.branching_weight(graph.copy())
+    nx.algorithms.tree.greedy_branching(graph.copy())
+
+    assert all(
+        "weight" not in d for (u, v, d) in graph.edges(data=True)
+    ), "The above calls should not modify the initial graph in-place"

rl/Lib/site-packages/networkx/algorithms/tree/tests/test_coding.py

@@ -0,0 +1,114 @@
+"""Unit tests for the :mod:`~networkx.algorithms.tree.coding` module."""
+
+from itertools import product
+
+import pytest
+
+import networkx as nx
+from networkx.utils import edges_equal, nodes_equal
+
+
+class TestPruferSequence:
+    """Unit tests for the Prüfer sequence encoding and decoding
+    functions.
+
+    """
+
+    def test_nontree(self):
+        with pytest.raises(nx.NotATree):
+            G = nx.cycle_graph(3)
+            nx.to_prufer_sequence(G)
+
+    def test_null_graph(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.to_prufer_sequence(nx.null_graph())
+
+    def test_trivial_graph(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.to_prufer_sequence(nx.trivial_graph())
+
+    def test_bad_integer_labels(self):
+        with pytest.raises(KeyError):
+            T = nx.Graph(nx.utils.pairwise("abc"))
+            nx.to_prufer_sequence(T)
+
+    def test_encoding(self):
+        """Tests for encoding a tree as a Prüfer sequence using the
+        iterative strategy.
+
+        """
+        # Example from Wikipedia.
+        tree = nx.Graph([(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)])
+        sequence = nx.to_prufer_sequence(tree)
+        assert sequence == [3, 3, 3, 4]
+
+    def test_decoding(self):
+        """Tests for decoding a tree from a Prüfer sequence."""
+        # Example from Wikipedia.
+        sequence = [3, 3, 3, 4]
+        tree = nx.from_prufer_sequence(sequence)
+        assert nodes_equal(list(tree), list(range(6)))
+        edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
+        assert edges_equal(list(tree.edges()), edges)
+
+    def test_decoding2(self):
+        # Example from "An Optimal Algorithm for Prufer Codes".
+        sequence = [2, 4, 0, 1, 3, 3]
+        tree = nx.from_prufer_sequence(sequence)
+        assert nodes_equal(list(tree), list(range(8)))
+        edges = [(0, 1), (0, 4), (1, 3), (2, 4), (2, 5), (3, 6), (3, 7)]
+        assert edges_equal(list(tree.edges()), edges)
+
+    def test_inverse(self):
+        """Tests that the encoding and decoding functions are inverses."""
+        for T in nx.nonisomorphic_trees(4):
+            T2 = nx.from_prufer_sequence(nx.to_prufer_sequence(T))
+            assert nodes_equal(list(T), list(T2))
+            assert edges_equal(list(T.edges()), list(T2.edges()))
+
+        for seq in product(range(4), repeat=2):
+            seq2 = nx.to_prufer_sequence(nx.from_prufer_sequence(seq))
+            assert list(seq) == seq2
+
+
+class TestNestedTuple:
+    """Unit tests for the nested tuple encoding and decoding functions."""
+
+    def test_nontree(self):
+        with pytest.raises(nx.NotATree):
+            G = nx.cycle_graph(3)
+            nx.to_nested_tuple(G, 0)
+
+    def test_unknown_root(self):
+        with pytest.raises(nx.NodeNotFound):
+            G = nx.path_graph(2)
+            nx.to_nested_tuple(G, "bogus")
+
+    def test_encoding(self):
+        T = nx.full_rary_tree(2, 2**3 - 1)
+        expected = (((), ()), ((), ()))
+        actual = nx.to_nested_tuple(T, 0)
+        assert nodes_equal(expected, actual)
+
+    def test_canonical_form(self):
+        T = nx.Graph()
+        T.add_edges_from([(0, 1), (0, 2), (0, 3)])
+        T.add_edges_from([(1, 4), (1, 5)])
+        T.add_edges_from([(3, 6), (3, 7)])
+        root = 0
+        actual = nx.to_nested_tuple(T, root, canonical_form=True)
+        expected = ((), ((), ()), ((), ()))
+        assert actual == expected
+
+    def test_decoding(self):
+        balanced = (((), ()), ((), ()))
+        expected = nx.full_rary_tree(2, 2**3 - 1)
+        actual = nx.from_nested_tuple(balanced)
+        assert nx.is_isomorphic(expected, actual)
+
+    def test_sensible_relabeling(self):
+        balanced = (((), ()), ((), ()))
+        T = nx.from_nested_tuple(balanced, sensible_relabeling=True)
+        edges = [(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)]
+        assert nodes_equal(list(T), list(range(2**3 - 1)))
+        assert edges_equal(list(T.edges()), edges)

rl/Lib/site-packages/networkx/algorithms/tree/tests/test_decomposition.py

@@ -0,0 +1,79 @@
+import networkx as nx
+from networkx.algorithms.tree.decomposition import junction_tree
+
+
+def test_junction_tree_directed_confounders():
+    B = nx.DiGraph()
+    B.add_edges_from([("A", "C"), ("B", "C"), ("C", "D"), ("C", "E")])
+
+    G = junction_tree(B)
+    J = nx.Graph()
+    J.add_edges_from(
+        [
+            (("C", "E"), ("C",)),
+            (("C",), ("A", "B", "C")),
+            (("A", "B", "C"), ("C",)),
+            (("C",), ("C", "D")),
+        ]
+    )
+
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_directed_unconnected_nodes():
+    B = nx.DiGraph()
+    B.add_nodes_from([("A", "B", "C", "D")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_nodes_from([("A", "B", "C", "D")])
+
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_directed_cascade():
+    B = nx.DiGraph()
+    B.add_edges_from([("A", "B"), ("B", "C"), ("C", "D")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_edges_from(
+        [
+            (("A", "B"), ("B",)),
+            (("B",), ("B", "C")),
+            (("B", "C"), ("C",)),
+            (("C",), ("C", "D")),
+        ]
+    )
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_directed_unconnected_edges():
+    B = nx.DiGraph()
+    B.add_edges_from([("A", "B"), ("C", "D"), ("E", "F")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_nodes_from([("A", "B"), ("C", "D"), ("E", "F")])
+
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_undirected():
+    B = nx.Graph()
+    B.add_edges_from([("A", "C"), ("A", "D"), ("B", "C"), ("C", "E")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_edges_from(
+        [
+            (("A", "D"), ("A",)),
+            (("A",), ("A", "C")),
+            (("A", "C"), ("C",)),
+            (("C",), ("B", "C")),
+            (("B", "C"), ("C",)),
+            (("C",), ("C", "E")),
+        ]
+    )
+
+    assert nx.is_isomorphic(G, J)

rl/Lib/site-packages/networkx/algorithms/tree/tests/test_mst.py

@@ -0,0 +1,918 @@
+"""Unit tests for the :mod:`networkx.algorithms.tree.mst` module."""
+
+import pytest
+
+import networkx as nx
+from networkx.utils import edges_equal, nodes_equal
+
+
+def test_unknown_algorithm():
+    with pytest.raises(ValueError):
+        nx.minimum_spanning_tree(nx.Graph(), algorithm="random")
+    with pytest.raises(
+        ValueError, match="random is not a valid choice for an algorithm."
+    ):
+        nx.maximum_spanning_edges(nx.Graph(), algorithm="random")
+
+
+class MinimumSpanningTreeTestBase:
+    """Base class for test classes for minimum spanning tree algorithms.
+    This class contains some common tests that will be inherited by
+    subclasses. Each subclass must have a class attribute
+    :data:`algorithm` that is a string representing the algorithm to
+    run, as described under the ``algorithm`` keyword argument for the
+    :func:`networkx.minimum_spanning_edges` function.  Subclasses can
+    then implement any algorithm-specific tests.
+    """
+
+    def setup_method(self, method):
+        """Creates an example graph and stores the expected minimum and
+        maximum spanning tree edges.
+        """
+        # This stores the class attribute `algorithm` in an instance attribute.
+        self.algo = self.algorithm
+        # This example graph comes from Wikipedia:
+        # https://en.wikipedia.org/wiki/Kruskal's_algorithm
+        edges = [
+            (0, 1, 7),
+            (0, 3, 5),
+            (1, 2, 8),
+            (1, 3, 9),
+            (1, 4, 7),
+            (2, 4, 5),
+            (3, 4, 15),
+            (3, 5, 6),
+            (4, 5, 8),
+            (4, 6, 9),
+            (5, 6, 11),
+        ]
+        self.G = nx.Graph()
+        self.G.add_weighted_edges_from(edges)
+        self.minimum_spanning_edgelist = [
+            (0, 1, {"weight": 7}),
+            (0, 3, {"weight": 5}),
+            (1, 4, {"weight": 7}),
+            (2, 4, {"weight": 5}),
+            (3, 5, {"weight": 6}),
+            (4, 6, {"weight": 9}),
+        ]
+        self.maximum_spanning_edgelist = [
+            (0, 1, {"weight": 7}),
+            (1, 2, {"weight": 8}),
+            (1, 3, {"weight": 9}),
+            (3, 4, {"weight": 15}),
+            (4, 6, {"weight": 9}),
+            (5, 6, {"weight": 11}),
+        ]
+
+    def test_minimum_edges(self):
+        edges = nx.minimum_spanning_edges(self.G, algorithm=self.algo)
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v), d) for u, v, d in edges)
+        assert edges_equal(actual, self.minimum_spanning_edgelist)
+
+    def test_maximum_edges(self):
+        edges = nx.maximum_spanning_edges(self.G, algorithm=self.algo)
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v), d) for u, v, d in edges)
+        assert edges_equal(actual, self.maximum_spanning_edgelist)
+
+    def test_without_data(self):
+        edges = nx.minimum_spanning_edges(self.G, algorithm=self.algo, data=False)
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        expected = [(u, v) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, expected)
+
+    def test_nan_weights(self):
+        # Edge weights NaN never appear in the spanning tree. see #2164
+        G = self.G
+        G.add_edge(0, 12, weight=float("nan"))
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=True
+        )
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        expected = [(u, v) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, expected)
+        # Now test for raising exception
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=False
+        )
+        with pytest.raises(ValueError):
+            list(edges)
+        # test default for ignore_nan as False
+        edges = nx.minimum_spanning_edges(G, algorithm=self.algo, data=False)
+        with pytest.raises(ValueError):
+            list(edges)
+
+    def test_nan_weights_MultiGraph(self):
+        G = nx.MultiGraph()
+        G.add_edge(0, 12, weight=float("nan"))
+        edges = nx.minimum_spanning_edges(
+            G, algorithm="prim", data=False, ignore_nan=False
+        )
+        with pytest.raises(ValueError):
+            list(edges)
+        # test default for ignore_nan as False
+        edges = nx.minimum_spanning_edges(G, algorithm="prim", data=False)
+        with pytest.raises(ValueError):
+            list(edges)
+
+    def test_nan_weights_order(self):
+        # now try again with a nan edge at the beginning of G.nodes
+        edges = [
+            (0, 1, 7),
+            (0, 3, 5),
+            (1, 2, 8),
+            (1, 3, 9),
+            (1, 4, 7),
+            (2, 4, 5),
+            (3, 4, 15),
+            (3, 5, 6),
+            (4, 5, 8),
+            (4, 6, 9),
+            (5, 6, 11),
+        ]
+        G = nx.Graph()
+        G.add_weighted_edges_from([(u + 1, v + 1, wt) for u, v, wt in edges])
+        G.add_edge(0, 7, weight=float("nan"))
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=True
+        )
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        shift = [(u + 1, v + 1) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, shift)
+
+    def test_isolated_node(self):
+        # now try again with an isolated node
+        edges = [
+            (0, 1, 7),
+            (0, 3, 5),
+            (1, 2, 8),
+            (1, 3, 9),
+            (1, 4, 7),
+            (2, 4, 5),
+            (3, 4, 15),
+            (3, 5, 6),
+            (4, 5, 8),
+            (4, 6, 9),
+            (5, 6, 11),
+        ]
+        G = nx.Graph()
+        G.add_weighted_edges_from([(u + 1, v + 1, wt) for u, v, wt in edges])
+        G.add_node(0)
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=True
+        )
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        shift = [(u + 1, v + 1) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, shift)
+
+    def test_minimum_tree(self):
+        T = nx.minimum_spanning_tree(self.G, algorithm=self.algo)
+        actual = sorted(T.edges(data=True))
+        assert edges_equal(actual, self.minimum_spanning_edgelist)
+
+    def test_maximum_tree(self):
+        T = nx.maximum_spanning_tree(self.G, algorithm=self.algo)
+        actual = sorted(T.edges(data=True))
+        assert edges_equal(actual, self.maximum_spanning_edgelist)
+
+    def test_disconnected(self):
+        G = nx.Graph([(0, 1, {"weight": 1}), (2, 3, {"weight": 2})])
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert nodes_equal(list(T), list(range(4)))
+        assert edges_equal(list(T.edges()), [(0, 1), (2, 3)])
+
+    def test_empty_graph(self):
+        G = nx.empty_graph(3)
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert nodes_equal(sorted(T), list(range(3)))
+        assert T.number_of_edges() == 0
+
+    def test_attributes(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1, color="red", distance=7)
+        G.add_edge(2, 3, weight=1, color="green", distance=2)
+        G.add_edge(1, 3, weight=10, color="blue", distance=1)
+        G.graph["foo"] = "bar"
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert T.graph == G.graph
+        assert nodes_equal(T, G)
+        for u, v in T.edges():
+            assert T.adj[u][v] == G.adj[u][v]
+
+    def test_weight_attribute(self):
+        G = nx.Graph()
+        G.add_edge(0, 1, weight=1, distance=7)
+        G.add_edge(0, 2, weight=30, distance=1)
+        G.add_edge(1, 2, weight=1, distance=1)
+        G.add_node(3)
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo, weight="distance")
+        assert nodes_equal(sorted(T), list(range(4)))
+        assert edges_equal(sorted(T.edges()), [(0, 2), (1, 2)])
+        T = nx.maximum_spanning_tree(G, algorithm=self.algo, weight="distance")
+        assert nodes_equal(sorted(T), list(range(4)))
+        assert edges_equal(sorted(T.edges()), [(0, 1), (0, 2)])
+
+
+class TestBoruvka(MinimumSpanningTreeTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Borůvka's algorithm.
+    """
+
+    algorithm = "boruvka"
+
+    def test_unicode_name(self):
+        """Tests that using a Unicode string can correctly indicate
+        Borůvka's algorithm.
+        """
+        edges = nx.minimum_spanning_edges(self.G, algorithm="borůvka")
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v), d) for u, v, d in edges)
+        assert edges_equal(actual, self.minimum_spanning_edgelist)
+
+
+class MultigraphMSTTestBase(MinimumSpanningTreeTestBase):
+    # Abstract class
+
+    def test_multigraph_keys_min(self):
+        """Tests that the minimum spanning edges of a multigraph
+        preserves edge keys.
+        """
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        min_edges = nx.minimum_spanning_edges
+        mst_edges = min_edges(G, algorithm=self.algo, data=False)
+        assert edges_equal([(0, 1, "b")], list(mst_edges))
+
+    def test_multigraph_keys_max(self):
+        """Tests that the maximum spanning edges of a multigraph
+        preserves edge keys.
+        """
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        max_edges = nx.maximum_spanning_edges
+        mst_edges = max_edges(G, algorithm=self.algo, data=False)
+        assert edges_equal([(0, 1, "a")], list(mst_edges))
+
+
+class TestKruskal(MultigraphMSTTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Kruskal's algorithm.
+    """
+
+    algorithm = "kruskal"
+
+    def test_key_data_bool(self):
+        """Tests that the keys and data values are included in
+        MST edges based on whether keys and data parameters are
+        true or false"""
+        G = nx.MultiGraph()
+        G.add_edge(1, 2, key=1, weight=2)
+        G.add_edge(1, 2, key=2, weight=3)
+        G.add_edge(3, 2, key=1, weight=2)
+        G.add_edge(3, 1, key=1, weight=4)
+
+        # keys are included and data is not included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=True, data=False
+        )
+        assert edges_equal([(1, 2, 1), (2, 3, 1)], list(mst_edges))
+
+        # keys are not included and data is included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=False, data=True
+        )
+        assert edges_equal(
+            [(1, 2, {"weight": 2}), (2, 3, {"weight": 2})], list(mst_edges)
+        )
+
+        # both keys and data are not included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=False, data=False
+        )
+        assert edges_equal([(1, 2), (2, 3)], list(mst_edges))
+
+        # both keys and data are included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=True, data=True
+        )
+        assert edges_equal(
+            [(1, 2, 1, {"weight": 2}), (2, 3, 1, {"weight": 2})], list(mst_edges)
+        )
+
+
+class TestPrim(MultigraphMSTTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Prim's algorithm.
+    """
+
+    algorithm = "prim"
+
+    def test_prim_mst_edges_simple_graph(self):
+        H = nx.Graph()
+        H.add_edge(1, 2, key=2, weight=3)
+        H.add_edge(3, 2, key=1, weight=2)
+        H.add_edge(3, 1, key=1, weight=4)
+
+        mst_edges = nx.minimum_spanning_edges(H, algorithm=self.algo, ignore_nan=True)
+        assert edges_equal(
+            [(1, 2, {"key": 2, "weight": 3}), (2, 3, {"key": 1, "weight": 2})],
+            list(mst_edges),
+        )
+
+    def test_ignore_nan(self):
+        """Tests that the edges with NaN weights are ignored or
+        raise an Error based on ignore_nan is true or false"""
+        H = nx.MultiGraph()
+        H.add_edge(1, 2, key=1, weight=float("nan"))
+        H.add_edge(1, 2, key=2, weight=3)
+        H.add_edge(3, 2, key=1, weight=2)
+        H.add_edge(3, 1, key=1, weight=4)
+
+        # NaN weight edges are ignored when ignore_nan=True
+        mst_edges = nx.minimum_spanning_edges(H, algorithm=self.algo, ignore_nan=True)
+        assert edges_equal(
+            [(1, 2, 2, {"weight": 3}), (2, 3, 1, {"weight": 2})], list(mst_edges)
+        )
+
+        # NaN weight edges raise Error when ignore_nan=False
+        with pytest.raises(ValueError):
+            list(nx.minimum_spanning_edges(H, algorithm=self.algo, ignore_nan=False))
+
+    def test_multigraph_keys_tree(self):
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert edges_equal([(0, 1, 1)], list(T.edges(data="weight")))
+
+    def test_multigraph_keys_tree_max(self):
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        T = nx.maximum_spanning_tree(G, algorithm=self.algo)
+        assert edges_equal([(0, 1, 2)], list(T.edges(data="weight")))
+
+
+class TestSpanningTreeIterator:
+    """
+    Tests the spanning tree iterator on the example graph in the 2005 Sörensen
+    and Janssens paper An Algorithm to Generate all Spanning Trees of a Graph in
+    Order of Increasing Cost
+    """
+
+    def setup_method(self):
+        # Original Graph
+        edges = [(0, 1, 5), (1, 2, 4), (1, 4, 6), (2, 3, 5), (2, 4, 7), (3, 4, 3)]
+        self.G = nx.Graph()
+        self.G.add_weighted_edges_from(edges)
+        # List of lists of spanning trees in increasing order
+        self.spanning_trees = [
+            # 1, MST, cost = 17
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (2, 3, {"weight": 5}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 2, cost = 18
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (1, 4, {"weight": 6}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 3, cost = 19
+            [
+                (0, 1, {"weight": 5}),
+                (1, 4, {"weight": 6}),
+                (2, 3, {"weight": 5}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 4, cost = 19
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (2, 4, {"weight": 7}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 5, cost = 20
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (1, 4, {"weight": 6}),
+                (2, 3, {"weight": 5}),
+            ],
+            # 6, cost = 21
+            [
+                (0, 1, {"weight": 5}),
+                (1, 4, {"weight": 6}),
+                (2, 4, {"weight": 7}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 7, cost = 21
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (2, 3, {"weight": 5}),
+                (2, 4, {"weight": 7}),
+            ],
+            # 8, cost = 23
+            [
+                (0, 1, {"weight": 5}),
+                (1, 4, {"weight": 6}),
+                (2, 3, {"weight": 5}),
+                (2, 4, {"weight": 7}),
+            ],
+        ]
+
+    def test_minimum_spanning_tree_iterator(self):
+        """
+        Tests that the spanning trees are correctly returned in increasing order
+        """
+        tree_index = 0
+        for tree in nx.SpanningTreeIterator(self.G):
+            actual = sorted(tree.edges(data=True))
+            assert edges_equal(actual, self.spanning_trees[tree_index])
+            tree_index += 1
+
+    def test_maximum_spanning_tree_iterator(self):
+        """
+        Tests that the spanning trees are correctly returned in decreasing order
+        """
+        tree_index = 7
+        for tree in nx.SpanningTreeIterator(self.G, minimum=False):
+            actual = sorted(tree.edges(data=True))
+            assert edges_equal(actual, self.spanning_trees[tree_index])
+            tree_index -= 1
+
+
+class TestSpanningTreeMultiGraphIterator:
+    """
+    Uses the same graph as the above class but with an added edge of twice the weight.
+    """
+
+    def setup_method(self):
+        # New graph
+        edges = [
+            (0, 1, 5),
+            (0, 1, 10),
+            (1, 2, 4),
+            (1, 2, 8),
+            (1, 4, 6),
+            (1, 4, 12),
+            (2, 3, 5),
+            (2, 3, 10),
+            (2, 4, 7),
+            (2, 4, 14),
+            (3, 4, 3),
+            (3, 4, 6),
+        ]
+        self.G = nx.MultiGraph()
+        self.G.add_weighted_edges_from(edges)
+
+        # There are 128 trees. I'd rather not list all 128 here, and computing them
+        # on such a small graph actually doesn't take that long.
+        from itertools import combinations
+
+        self.spanning_trees = []
+        for e in combinations(self.G.edges, 4):
+            tree = self.G.edge_subgraph(e)
+            if nx.is_tree(tree):
+                self.spanning_trees.append(sorted(tree.edges(keys=True, data=True)))
+
+    def test_minimum_spanning_tree_iterator_multigraph(self):
+        """
+        Tests that the spanning trees are correctly returned in increasing order
+        """
+        tree_index = 0
+        last_weight = 0
+        for tree in nx.SpanningTreeIterator(self.G):
+            actual = sorted(tree.edges(keys=True, data=True))
+            weight = sum([e[3]["weight"] for e in actual])
+            assert actual in self.spanning_trees
+            assert weight >= last_weight
+            tree_index += 1
+
+    def test_maximum_spanning_tree_iterator_multigraph(self):
+        """
+        Tests that the spanning trees are correctly returned in decreasing order
+        """
+        tree_index = 127
+        # Maximum weight tree is 46
+        last_weight = 50
+        for tree in nx.SpanningTreeIterator(self.G, minimum=False):
+            actual = sorted(tree.edges(keys=True, data=True))
+            weight = sum([e[3]["weight"] for e in actual])
+            assert actual in self.spanning_trees
+            assert weight <= last_weight
+            tree_index -= 1
+
+
+def test_random_spanning_tree_multiplicative_small():
+    """
+    Using a fixed seed, sample one tree for repeatability.
+    """
+    from math import exp
+
+    pytest.importorskip("scipy")
+
+    gamma = {
+        (0, 1): -0.6383,
+        (0, 2): -0.6827,
+        (0, 5): 0,
+        (1, 2): -1.0781,
+        (1, 4): 0,
+        (2, 3): 0,
+        (5, 3): -0.2820,
+        (5, 4): -0.3327,
+        (4, 3): -0.9927,
+    }
+
+    # The undirected support of gamma
+    G = nx.Graph()
+    for u, v in gamma:
+        G.add_edge(u, v, lambda_key=exp(gamma[(u, v)]))
+
+    solution_edges = [(2, 3), (3, 4), (0, 5), (5, 4), (4, 1)]
+    solution = nx.Graph()
+    solution.add_edges_from(solution_edges)
+
+    sampled_tree = nx.random_spanning_tree(G, "lambda_key", seed=42)
+
+    assert nx.utils.edges_equal(solution.edges, sampled_tree.edges)
+
+
+@pytest.mark.slow
+def test_random_spanning_tree_multiplicative_large():
+    """
+    Sample many trees from the distribution created in the last test
+    """
+    from math import exp
+    from random import Random
+
+    pytest.importorskip("numpy")
+    stats = pytest.importorskip("scipy.stats")
+
+    gamma = {
+        (0, 1): -0.6383,
+        (0, 2): -0.6827,
+        (0, 5): 0,
+        (1, 2): -1.0781,
+        (1, 4): 0,
+        (2, 3): 0,
+        (5, 3): -0.2820,
+        (5, 4): -0.3327,
+        (4, 3): -0.9927,
+    }
+
+    # The undirected support of gamma
+    G = nx.Graph()
+    for u, v in gamma:
+        G.add_edge(u, v, lambda_key=exp(gamma[(u, v)]))
+
+    # Find the multiplicative weight for each tree.
+    total_weight = 0
+    tree_expected = {}
+    for t in nx.SpanningTreeIterator(G):
+        # Find the multiplicative weight of the spanning tree
+        weight = 1
+        for u, v, d in t.edges(data="lambda_key"):
+            weight *= d
+        tree_expected[t] = weight
+        total_weight += weight
+
+    # Assert that every tree has an entry in the expected distribution
+    assert len(tree_expected) == 75
+
+    # Set the sample size and then calculate the expected number of times we
+    # expect to see each tree. This test uses a near minimum sample size where
+    # the most unlikely tree has an expected frequency of 5.15.
+    # (Minimum required is 5)
+    #
+    # Here we also initialize the tree_actual dict so that we know the keys
+    # match between the two. We will later take advantage of the fact that since
+    # python 3.7 dict order is guaranteed so the expected and actual data will
+    # have the same order.
+    sample_size = 1200
+    tree_actual = {}
+    for t in tree_expected:
+        tree_expected[t] = (tree_expected[t] / total_weight) * sample_size
+        tree_actual[t] = 0
+
+    # Sample the spanning trees
+    #
+    # Assert that they are actually trees and record which of the 75 trees we
+    # have sampled.
+    #
+    # For repeatability, we want to take advantage of the decorators in NetworkX
+    # to randomly sample the same sample each time. However, if we pass in a
+    # constant seed to sample_spanning_tree we will get the same tree each time.
+    # Instead, we can create our own random number generator with a fixed seed
+    # and pass those into sample_spanning_tree.
+    rng = Random(37)
+    for _ in range(sample_size):
+        sampled_tree = nx.random_spanning_tree(G, "lambda_key", seed=rng)
+        assert nx.is_tree(sampled_tree)
+
+        for t in tree_expected:
+            if nx.utils.edges_equal(t.edges, sampled_tree.edges):
+                tree_actual[t] += 1
+                break
+
+    # Conduct a Chi squared test to see if the actual distribution matches the
+    # expected one at an alpha = 0.05 significance level.
+    #
+    # H_0: The distribution of trees in tree_actual matches the normalized product
+    # of the edge weights in the tree.
+    #
+    # H_a: The distribution of trees in tree_actual follows some other
+    # distribution of spanning trees.
+    _, p = stats.chisquare(list(tree_actual.values()), list(tree_expected.values()))
+
+    # Assert that p is greater than the significance level so that we do not
+    # reject the null hypothesis
+    assert not p < 0.05
+
+
+def test_random_spanning_tree_additive_small():
+    """
+    Sample a single spanning tree from the additive method.
+    """
+    pytest.importorskip("scipy")
+
+    edges = {
+        (0, 1): 1,
+        (0, 2): 1,
+        (0, 5): 3,
+        (1, 2): 2,
+        (1, 4): 3,
+        (2, 3): 3,
+        (5, 3): 4,
+        (5, 4): 5,
+        (4, 3): 4,
+    }
+
+    # Build the graph
+    G = nx.Graph()
+    for u, v in edges:
+        G.add_edge(u, v, weight=edges[(u, v)])
+
+    solution_edges = [(0, 2), (1, 2), (2, 3), (3, 4), (3, 5)]
+    solution = nx.Graph()
+    solution.add_edges_from(solution_edges)
+
+    sampled_tree = nx.random_spanning_tree(
+        G, weight="weight", multiplicative=False, seed=37
+    )
+
+    assert nx.utils.edges_equal(solution.edges, sampled_tree.edges)
+
+
+@pytest.mark.slow
+def test_random_spanning_tree_additive_large():
+    """
+    Sample many spanning trees from the additive method.
+    """
+    from random import Random
+
+    pytest.importorskip("numpy")
+    stats = pytest.importorskip("scipy.stats")
+
+    edges = {
+        (0, 1): 1,
+        (0, 2): 1,
+        (0, 5): 3,
+        (1, 2): 2,
+        (1, 4): 3,
+        (2, 3): 3,
+        (5, 3): 4,
+        (5, 4): 5,
+        (4, 3): 4,
+    }
+
+    # Build the graph
+    G = nx.Graph()
+    for u, v in edges:
+        G.add_edge(u, v, weight=edges[(u, v)])
+
+    # Find the additive weight for each tree.
+    total_weight = 0
+    tree_expected = {}
+    for t in nx.SpanningTreeIterator(G):
+        # Find the multiplicative weight of the spanning tree
+        weight = 0
+        for u, v, d in t.edges(data="weight"):
+            weight += d
+        tree_expected[t] = weight
+        total_weight += weight
+
+    # Assert that every tree has an entry in the expected distribution
+    assert len(tree_expected) == 75
+
+    # Set the sample size and then calculate the expected number of times we
+    # expect to see each tree. This test uses a near minimum sample size where
+    # the most unlikely tree has an expected frequency of 5.07.
+    # (Minimum required is 5)
+    #
+    # Here we also initialize the tree_actual dict so that we know the keys
+    # match between the two. We will later take advantage of the fact that since
+    # python 3.7 dict order is guaranteed so the expected and actual data will
+    # have the same order.
+    sample_size = 500
+    tree_actual = {}
+    for t in tree_expected:
+        tree_expected[t] = (tree_expected[t] / total_weight) * sample_size
+        tree_actual[t] = 0
+
+    # Sample the spanning trees
+    #
+    # Assert that they are actually trees and record which of the 75 trees we
+    # have sampled.
+    #
+    # For repeatability, we want to take advantage of the decorators in NetworkX
+    # to randomly sample the same sample each time. However, if we pass in a
+    # constant seed to sample_spanning_tree we will get the same tree each time.
+    # Instead, we can create our own random number generator with a fixed seed
+    # and pass those into sample_spanning_tree.
+    rng = Random(37)
+    for _ in range(sample_size):
+        sampled_tree = nx.random_spanning_tree(
+            G, "weight", multiplicative=False, seed=rng
+        )
+        assert nx.is_tree(sampled_tree)
+
+        for t in tree_expected:
+            if nx.utils.edges_equal(t.edges, sampled_tree.edges):
+                tree_actual[t] += 1
+                break
+
+    # Conduct a Chi squared test to see if the actual distribution matches the
+    # expected one at an alpha = 0.05 significance level.
+    #
+    # H_0: The distribution of trees in tree_actual matches the normalized product
+    # of the edge weights in the tree.
+    #
+    # H_a: The distribution of trees in tree_actual follows some other
+    # distribution of spanning trees.
+    _, p = stats.chisquare(list(tree_actual.values()), list(tree_expected.values()))
+
+    # Assert that p is greater than the significance level so that we do not
+    # reject the null hypothesis
+    assert not p < 0.05
+
+
+def test_random_spanning_tree_empty_graph():
+    G = nx.Graph()
+    rst = nx.tree.random_spanning_tree(G)
+    assert len(rst.nodes) == 0
+    assert len(rst.edges) == 0
+
+
+def test_random_spanning_tree_single_node_graph():
+    G = nx.Graph()
+    G.add_node(0)
+    rst = nx.tree.random_spanning_tree(G)
+    assert len(rst.nodes) == 1
+    assert len(rst.edges) == 0
+
+
+def test_random_spanning_tree_single_node_loop():
+    G = nx.Graph()
+    G.add_node(0)
+    G.add_edge(0, 0)
+    rst = nx.tree.random_spanning_tree(G)
+    assert len(rst.nodes) == 1
+    assert len(rst.edges) == 0
+
+
+class TestNumberSpanningTrees:
+    @classmethod
+    def setup_class(cls):
+        global np
+        np = pytest.importorskip("numpy")
+        sp = pytest.importorskip("scipy")
+
+    def test_nst_disconnected(self):
+        G = nx.empty_graph(2)
+        assert np.isclose(nx.number_of_spanning_trees(G), 0)
+
+    def test_nst_no_nodes(self):
+        G = nx.Graph()
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.number_of_spanning_trees(G)
+
+    def test_nst_weight(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1)
+        G.add_edge(1, 3, weight=1)
+        G.add_edge(2, 3, weight=2)
+        # weights are ignored
+        assert np.isclose(nx.number_of_spanning_trees(G), 3)
+        # including weight
+        assert np.isclose(nx.number_of_spanning_trees(G, weight="weight"), 5)
+
+    def test_nst_negative_weight(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1)
+        G.add_edge(1, 3, weight=-1)
+        G.add_edge(2, 3, weight=-2)
+        # weights are ignored
+        assert np.isclose(nx.number_of_spanning_trees(G), 3)
+        # including weight
+        assert np.isclose(nx.number_of_spanning_trees(G, weight="weight"), -1)
+
+    def test_nst_selfloop(self):
+        # self-loops are ignored
+        G = nx.complete_graph(3)
+        G.add_edge(1, 1)
+        assert np.isclose(nx.number_of_spanning_trees(G), 3)
+
+    def test_nst_multigraph(self):
+        G = nx.MultiGraph()
+        G.add_edge(1, 2)
+        G.add_edge(1, 2)
+        G.add_edge(1, 3)
+        G.add_edge(2, 3)
+        assert np.isclose(nx.number_of_spanning_trees(G), 5)
+
+    def test_nst_complete_graph(self):
+        # this is known as Cayley's formula
+        N = 5
+        G = nx.complete_graph(N)
+        assert np.isclose(nx.number_of_spanning_trees(G), N ** (N - 2))
+
+    def test_nst_path_graph(self):
+        G = nx.path_graph(5)
+        assert np.isclose(nx.number_of_spanning_trees(G), 1)
+
+    def test_nst_cycle_graph(self):
+        G = nx.cycle_graph(5)
+        assert np.isclose(nx.number_of_spanning_trees(G), 5)
+
+    def test_nst_directed_noroot(self):
+        G = nx.empty_graph(3, create_using=nx.MultiDiGraph)
+        with pytest.raises(nx.NetworkXError):
+            nx.number_of_spanning_trees(G)
+
+    def test_nst_directed_root_not_exist(self):
+        G = nx.empty_graph(3, create_using=nx.MultiDiGraph)
+        with pytest.raises(nx.NetworkXError):
+            nx.number_of_spanning_trees(G, root=42)
+
+    def test_nst_directed_not_weak_connected(self):
+        G = nx.DiGraph()
+        G.add_edge(1, 2)
+        G.add_edge(3, 4)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=1), 0)
+
+    def test_nst_directed_cycle_graph(self):
+        G = nx.DiGraph()
+        G = nx.cycle_graph(7, G)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 1)
+
+    def test_nst_directed_complete_graph(self):
+        G = nx.DiGraph()
+        G = nx.complete_graph(7, G)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 7**5)
+
+    def test_nst_directed_multi(self):
+        G = nx.MultiDiGraph()
+        G = nx.cycle_graph(3, G)
+        G.add_edge(1, 2)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 2)
+
+    def test_nst_directed_selfloop(self):
+        G = nx.MultiDiGraph()
+        G = nx.cycle_graph(3, G)
+        G.add_edge(1, 1)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 1)
+
+    def test_nst_directed_weak_connected(self):
+        G = nx.MultiDiGraph()
+        G = nx.cycle_graph(3, G)
+        G.remove_edge(1, 2)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 0)
+
+    def test_nst_directed_weighted(self):
+        # from root=1:
+        # arborescence 1: 1->2, 1->3, weight=2*1
+        # arborescence 2: 1->2, 2->3, weight=2*3
+        G = nx.DiGraph()
+        G.add_edge(1, 2, weight=2)
+        G.add_edge(1, 3, weight=1)
+        G.add_edge(2, 3, weight=3)
+        Nst = nx.number_of_spanning_trees(G, root=1, weight="weight")
+        assert np.isclose(Nst, 8)
+        Nst = nx.number_of_spanning_trees(G, root=2, weight="weight")
+        assert np.isclose(Nst, 0)
+        Nst = nx.number_of_spanning_trees(G, root=3, weight="weight")
+        assert np.isclose(Nst, 0)

rl/Lib/site-packages/networkx/algorithms/tree/tests/test_operations.py

@@ -0,0 +1,53 @@
+from itertools import chain
+
+import networkx as nx
+from networkx.utils import edges_equal, nodes_equal
+
+
+def _check_custom_label_attribute(input_trees, res_tree, label_attribute):
+    res_attr_dict = nx.get_node_attributes(res_tree, label_attribute)
+    res_attr_set = set(res_attr_dict.values())
+    input_label = (tree for tree, root in input_trees)
+    input_label_set = set(chain.from_iterable(input_label))
+    return res_attr_set == input_label_set
+
+
+def test_empty_sequence():
+    """Joining the empty sequence results in the tree with one node."""
+    T = nx.join_trees([])
+    assert len(T) == 1
+    assert T.number_of_edges() == 0
+
+
+def test_single():
+    """Joining just one tree yields a tree with one more node."""
+    T = nx.empty_graph(1)
+    trees = [(T, 0)]
+    actual_with_label = nx.join_trees(trees, label_attribute="custom_label")
+    expected = nx.path_graph(2)
+    assert nodes_equal(list(expected), list(actual_with_label))
+    assert edges_equal(list(expected.edges()), list(actual_with_label.edges()))
+
+
+def test_basic():
+    """Joining multiple subtrees at a root node."""
+    trees = [(nx.full_rary_tree(2, 2**2 - 1), 0) for i in range(2)]
+    expected = nx.full_rary_tree(2, 2**3 - 1)
+    actual = nx.join_trees(trees, label_attribute="old_labels")
+    assert nx.is_isomorphic(actual, expected)
+    assert _check_custom_label_attribute(trees, actual, "old_labels")
+
+    actual_without_label = nx.join_trees(trees)
+    assert nx.is_isomorphic(actual_without_label, expected)
+    # check that no labels were stored
+    assert all(not data for _, data in actual_without_label.nodes(data=True))
+
+
+def test_first_label():
+    """Test the functionality of the first_label argument."""
+    T1 = nx.path_graph(3)
+    T2 = nx.path_graph(2)
+    actual = nx.join_trees([(T1, 0), (T2, 0)], first_label=10)
+    expected_nodes = set(range(10, 16))
+    assert set(actual.nodes()) == expected_nodes
+    assert set(actual.neighbors(10)) == {11, 14}

rl/Lib/site-packages/networkx/algorithms/tree/tests/test_recognition.py

@@ -0,0 +1,174 @@
+import pytest
+
+import networkx as nx
+
+
+class TestTreeRecognition:
+    graph = nx.Graph
+    multigraph = nx.MultiGraph
+
+    @classmethod
+    def setup_class(cls):
+        cls.T1 = cls.graph()
+
+        cls.T2 = cls.graph()
+        cls.T2.add_node(1)
+
+        cls.T3 = cls.graph()
+        cls.T3.add_nodes_from(range(5))
+        edges = [(i, i + 1) for i in range(4)]
+        cls.T3.add_edges_from(edges)
+
+        cls.T5 = cls.multigraph()
+        cls.T5.add_nodes_from(range(5))
+        edges = [(i, i + 1) for i in range(4)]
+        cls.T5.add_edges_from(edges)
+
+        cls.T6 = cls.graph()
+        cls.T6.add_nodes_from([6, 7])
+        cls.T6.add_edge(6, 7)
+
+        cls.F1 = nx.compose(cls.T6, cls.T3)
+
+        cls.N4 = cls.graph()
+        cls.N4.add_node(1)
+        cls.N4.add_edge(1, 1)
+
+        cls.N5 = cls.graph()
+        cls.N5.add_nodes_from(range(5))
+
+        cls.N6 = cls.graph()
+        cls.N6.add_nodes_from(range(3))
+        cls.N6.add_edges_from([(0, 1), (1, 2), (2, 0)])
+
+        cls.NF1 = nx.compose(cls.T6, cls.N6)
+
+    def test_null_tree(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_tree(self.graph())
+
+    def test_null_tree2(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_tree(self.multigraph())
+
+    def test_null_forest(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_forest(self.graph())
+
+    def test_null_forest2(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_forest(self.multigraph())
+
+    def test_is_tree(self):
+        assert nx.is_tree(self.T2)
+        assert nx.is_tree(self.T3)
+        assert nx.is_tree(self.T5)
+
+    def test_is_not_tree(self):
+        assert not nx.is_tree(self.N4)
+        assert not nx.is_tree(self.N5)
+        assert not nx.is_tree(self.N6)
+
+    def test_is_forest(self):
+        assert nx.is_forest(self.T2)
+        assert nx.is_forest(self.T3)
+        assert nx.is_forest(self.T5)
+        assert nx.is_forest(self.F1)
+        assert nx.is_forest(self.N5)
+
+    def test_is_not_forest(self):
+        assert not nx.is_forest(self.N4)
+        assert not nx.is_forest(self.N6)
+        assert not nx.is_forest(self.NF1)
+
+
+class TestDirectedTreeRecognition(TestTreeRecognition):
+    graph = nx.DiGraph
+    multigraph = nx.MultiDiGraph
+
+
+def test_disconnected_graph():
+    # https://github.com/networkx/networkx/issues/1144
+    G = nx.Graph()
+    G.add_edges_from([(0, 1), (1, 2), (2, 0), (3, 4)])
+    assert not nx.is_tree(G)
+
+    G = nx.DiGraph()
+    G.add_edges_from([(0, 1), (1, 2), (2, 0), (3, 4)])
+    assert not nx.is_tree(G)
+
+
+def test_dag_nontree():
+    G = nx.DiGraph()
+    G.add_edges_from([(0, 1), (0, 2), (1, 2)])
+    assert not nx.is_tree(G)
+    assert nx.is_directed_acyclic_graph(G)
+
+
+def test_multicycle():
+    G = nx.MultiDiGraph()
+    G.add_edges_from([(0, 1), (0, 1)])
+    assert not nx.is_tree(G)
+    assert nx.is_directed_acyclic_graph(G)
+
+
+def test_emptybranch():
+    G = nx.DiGraph()
+    G.add_nodes_from(range(10))
+    assert nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_is_branching_empty_graph_raises():
+    G = nx.DiGraph()
+    with pytest.raises(nx.NetworkXPointlessConcept, match="G has no nodes."):
+        nx.is_branching(G)
+
+
+def test_path():
+    G = nx.DiGraph()
+    nx.add_path(G, range(5))
+    assert nx.is_branching(G)
+    assert nx.is_arborescence(G)
+
+
+def test_notbranching1():
+    # Acyclic violation.
+    G = nx.MultiDiGraph()
+    G.add_nodes_from(range(10))
+    G.add_edges_from([(0, 1), (1, 0)])
+    assert not nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_notbranching2():
+    # In-degree violation.
+    G = nx.MultiDiGraph()
+    G.add_nodes_from(range(10))
+    G.add_edges_from([(0, 1), (0, 2), (3, 2)])
+    assert not nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_notarborescence1():
+    # Not an arborescence due to not spanning.
+    G = nx.MultiDiGraph()
+    G.add_nodes_from(range(10))
+    G.add_edges_from([(0, 1), (0, 2), (1, 3), (5, 6)])
+    assert nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_notarborescence2():
+    # Not an arborescence due to in-degree violation.
+    G = nx.MultiDiGraph()
+    nx.add_path(G, range(5))
+    G.add_edge(6, 4)
+    assert not nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_is_arborescense_empty_graph_raises():
+    G = nx.DiGraph()
+    with pytest.raises(nx.NetworkXPointlessConcept, match="G has no nodes."):
+        nx.is_arborescence(G)