Fast Generation of Unlabelled Free Trees using Weight Sequences

Abstract

In this paper, we introduce a new representation for ordered trees, the weight sequence representation. We then use this to construct new representations for both rooted trees and free trees, namely the canonical weight sequence representation. We construct algorithms for generating the weight sequence representations for all rooted and free trees of order $n$, and then add a number of modifications to improve the efficiency of the algorithms. Python implementations of the algorithms incorporate further improvements by using generators to avoid having to store the long lists of trees returned by the recursive calls, as well as caching the lists for rooted trees of small order, thereby eliminating many of the recursive calls. We further show how the algorithm can be modified to generate adjacency list and adjacency matrix representations for free trees. We compared the runtimes of our Python implementation for generating free trees with the Python implementation of the well-known WROM algorithm taken from NetworkX. The implementation of our algorithm is over four times as fast as the implementation of the WROM algorithm. The runtimes for generating adjacency lists and matrices are somewhat longer than those for weight sequences, but are still over four times as fast as the corresponding implementations of the WROM algorithm.