How to count trees?

TL;DR

The paper introduces a novel topological invariant for unlabeled trees that helps estimate the number of distinct tree structures up to 17 nodes, surpassing previous asymptotic bounds.

Contribution

A new invariant based on Nx2 matrices of node degrees and distances, enabling improved enumeration of unlabeled trees.

Findings

Invariant slightly exceeds Otter's asymptotic estimate

Allows lower bounds for the number of unlabeled trees

Effective for trees up to 17 nodes

Abstract

We propose a new topological invariant of unlabeled trees of N nodes. The invariant is a set of Nx2 matrices of integers, with sum_j k^{d_{i,j}} and v_i as the matrix elements, where d_{i,j} are the elements of the distance matrix and v_i denotes i-th node's degree and k in N. To compare the invariant calculated for possibly different graphs, the matrix rows are ordered with respect to first column, and -- if necessary -- with respect to the second one. We use the new invariant to evaluate from below the number of topologically different unlabeled trees up to N=17. The results slightly exceed the asymptotic evaluation of Otter.