Fixed database typo and removed unnecessary class identifier.

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

@@ -0,0 +1,451 @@
+import pytest
+
+np = pytest.importorskip("numpy")
+
+import networkx as nx
+
+
+from networkx.algorithms.tree import branchings
+from networkx.algorithms.tree import 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)
+# 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):
+    G = nx.DiGraph()
+    for u, v, weight in edges:
+        G.add_edge(u, v, 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)
+
+    # If we have an exception, let's see the edges.
+    print(e1)
+    print(e2)
+    print
+
+    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_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_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)
+
+    ed = branchings.Edmonds(G)
+    B = ed.find_optimum("weight", preserve_attrs=True, seed=1)
+
+    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
+
+    ed = branchings.Edmonds(G)
+    B = ed.find_optimum("weight", 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)
+
+    ed = branchings.Edmonds(G)
+    B = ed.find_optimum("weight", preserve_attrs=False)
+
+    edge_dict = B[0][1]
+    with pytest.raises(KeyError):
+        _ = edge_dict["otherattr"]

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

@@ -0,0 +1,117 @@
+"""Unit tests for the :mod:`~networkx.algorithms.tree.coding` module."""
+from itertools import product
+
+import pytest
+import networkx as nx
+from networkx.testing import assert_nodes_equal
+from networkx.testing import assert_edges_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)

venv/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)

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

@@ -0,0 +1,285 @@
+"""Unit tests for the :mod:`networkx.algorithms.tree.mst` module."""
+
+import pytest
+
+import networkx as nx
+from networkx.testing import assert_nodes_equal, assert_edges_equal
+
+
+def test_unknown_algorithm():
+    with pytest.raises(ValueError):
+        nx.minimum_spanning_tree(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_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, dict(weight=1)), (2, 3, dict(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"
+
+
+class TestPrim(MultigraphMSTTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Prim's algorithm.
+
+    """
+
+    algorithm = "prim"
+
+    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)
+        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)
+        assert_edges_equal([(0, 1, 2)], list(T.edges(data="weight")))

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

@@ -0,0 +1,38 @@
+"""Unit tests for the :mod:`networkx.algorithms.tree.operations` module.
+
+"""
+
+import networkx as nx
+from networkx.testing import assert_nodes_equal
+from networkx.testing import assert_edges_equal
+
+
+class TestJoin:
+    """Unit tests for the :func:`networkx.tree.join` function."""
+
+    def test_empty_sequence(self):
+        """Tests that joining the empty sequence results in the tree
+        with one node.
+
+        """
+        T = nx.join([])
+        assert len(T) == 1
+        assert T.number_of_edges() == 0
+
+    def test_single(self):
+        """Tests that joining just one tree yields a tree with one more
+        node.
+
+        """
+        T = nx.empty_graph(1)
+        actual = nx.join([(T, 0)])
+        expected = nx.path_graph(2)
+        assert_nodes_equal(list(expected), list(actual))
+        assert_edges_equal(list(expected.edges()), list(actual.edges()))
+
+    def test_basic(self):
+        """Tests for joining multiple subtrees at a root node."""
+        trees = [(nx.full_rary_tree(2, 2 ** 2 - 1), 0) for i in range(2)]
+        actual = nx.join(trees)
+        expected = nx.full_rary_tree(2, 2 ** 3 - 1)
+        assert nx.is_isomorphic(actual, expected)

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

@@ -0,0 +1,163 @@
+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_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)