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@hagberg hagberg/wrom.py
Created Dec 15, 2013

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generate non-isomorphic free trees or rooted trees
#!/usr/bin/env python
"""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."""
import sys
def trees(order):
"""Generate all the non-isomorphic free trees of the given
order."""
# start at the path graph rooted at its center
layout = range(order / 2 + 1) + range(1, (order + 1) / 2)
while layout is not None:
layout = next_tree(layout)
if layout != None:
yield layout_to_matrix(layout)
layout = next_rooted_tree(layout)
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):
"""Return 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
if __name__ == "__main__":
# when run from the command line,
# count the free trees of order argv[1]
import functools
generator = trees(int(sys.argv[1]))
print functools.reduce(lambda accum, tree: accum + 1,
generator, 0)
# vim:sw=4:sts=4:tw=72:fo=cq:
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