# How to count trees?

Source: [How to count trees?](<https://www.sota2.com/research/paper/68fa200a7a2b558e2932e35e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-23T20:36:06.504Z

Authors: S.Piec, K.Malarz, K.Kulakowski

Published: 2005-01-25T13:41:50.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/cond-mat/0501594>)

[PDF](<https://arxiv.org/pdf/cond-mat/0501594.pdf>)