"""
Implementation of the Wright, Richmond, Odlyzko and McKay (WROM)
algorithm for the enumeration of all non-isomorphic free trees of a
given order. Rooted trees are represented by level sequences, i.e.,
lists in which the i-th element specifies the distance of vertex i to
the root.
"""
__all__ = ["nonisomorphic_trees", "number_of_nonisomorphic_trees"]
import networkx as nx
[docs]def nonisomorphic_trees(order, create="graph"):
"""Returns a list of nonisomporphic trees
Parameters
----------
order : int
order of the desired tree(s)
create : graph or matrix (default="Graph)
If graph is selected a list of trees will be returned,
if matrix is selected a list of adjancency matrix will
be returned
Returns
-------
G : List of NetworkX Graphs
M : List of Adjacency matrices
References
----------
"""
if order < 2:
raise ValueError
# start at the path graph rooted at its center
layout = list(range(order // 2 + 1)) + list(range(1, (order + 1) // 2))
while layout is not None:
layout = _next_tree(layout)
if layout is not None:
if create == "graph":
yield _layout_to_graph(layout)
elif create == "matrix":
yield _layout_to_matrix(layout)
layout = _next_rooted_tree(layout)
[docs]def number_of_nonisomorphic_trees(order):
"""Returns the number of nonisomorphic trees
Parameters
----------
order : int
order of the desired tree(s)
Returns
-------
length : Number of nonisomorphic graphs for the given order
References
----------
"""
return sum(1 for _ in nonisomorphic_trees(order))
def _next_rooted_tree(predecessor, p=None):
"""One iteration of the Beyer-Hedetniemi algorithm."""
if p is None:
p = len(predecessor) - 1
while predecessor[p] == 1:
p -= 1
if p == 0:
return None
q = p - 1
while predecessor[q] != predecessor[p] - 1:
q -= 1
result = list(predecessor)
for i in range(p, len(result)):
result[i] = result[i - p + q]
return result
def _next_tree(candidate):
"""One iteration of the Wright, Richmond, Odlyzko and McKay
algorithm."""
# valid representation of a free tree if:
# there are at least two vertices at layer 1
# (this is always the case because we start at the path graph)
left, rest = _split_tree(candidate)
# and the left subtree of the root
# is less high than the tree with the left subtree removed
left_height = max(left)
rest_height = max(rest)
valid = rest_height >= left_height
if valid and rest_height == left_height:
# and, if left and rest are of the same height,
# if left does not encompass more vertices
if len(left) > len(rest):
valid = False
# and, if they have the same number or vertices,
# if left does not come after rest lexicographically
elif len(left) == len(rest) and left > rest:
valid = False
if valid:
return candidate
else:
# jump to the next valid free tree
p = len(left)
new_candidate = _next_rooted_tree(candidate, p)
if candidate[p] > 2:
new_left, new_rest = _split_tree(new_candidate)
new_left_height = max(new_left)
suffix = range(1, new_left_height + 2)
new_candidate[-len(suffix) :] = suffix
return new_candidate
def _split_tree(layout):
"""Returns a tuple of two layouts, one containing the left
subtree of the root vertex, and one containing the original tree
with the left subtree removed."""
one_found = False
m = None
for i in range(len(layout)):
if layout[i] == 1:
if one_found:
m = i
break
else:
one_found = True
if m is None:
m = len(layout)
left = [layout[i] - 1 for i in range(1, m)]
rest = [0] + [layout[i] for i in range(m, len(layout))]
return (left, rest)
def _layout_to_matrix(layout):
"""Create the adjacency matrix for the tree specified by the
given layout (level sequence)."""
result = [[0] * len(layout) for i in range(len(layout))]
stack = []
for i in range(len(layout)):
i_level = layout[i]
if stack:
j = stack[-1]
j_level = layout[j]
while j_level >= i_level:
stack.pop()
j = stack[-1]
j_level = layout[j]
result[i][j] = result[j][i] = 1
stack.append(i)
return result
def _layout_to_graph(layout):
"""Create a NetworkX Graph for the tree specified by the
given layout(level sequence)"""
result = [[0] * len(layout) for i in range(len(layout))]
G = nx.Graph()
stack = []
for i in range(len(layout)):
i_level = layout[i]
if stack:
j = stack[-1]
j_level = layout[j]
while j_level >= i_level:
stack.pop()
j = stack[-1]
j_level = layout[j]
G.add_edge(i, j)
stack.append(i)
return G