On symmetries in phylogenetic trees

TL;DR

This paper provides a combinatorial proof for a formula counting leaf-labelled rooted binary trees fixed by permutations, simplifying the understanding and random generation of tangled chains in phylogenetics.

Contribution

It offers a combinatorial proof of an existing formula for counting symmetric phylogenetic trees, improving the random sampling process for tangled chains.

Findings

Confirmed the formula for fixed phylogenetic trees under permutations.

Simplified the random sampling procedure for tangled chains.

Enhanced understanding of symmetries in phylogenetic trees.

Abstract

Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\geq 1$ leaves, fixed (for the relabelling action) by a given permutation $\sigma\in\frak{S}_n$. Denoting by $\lambda\vdash n$ the integer partition giving the sizes of the cycles of $\sigma$ in non-increasing order, they show by a guessing/checking approach that if $\lambda$ is a binary partition (it is known that $a_n(\sigma)=0$ otherwise), then $$ a_n(\sigma)=\prod_{i=2}^{\ell(\lambda)}(2(\lambda_i+\cdots+\lambda_{\ell(\lambda)})-1), $$ and they derive from it a formula and random generation procedure for tanglegrams (and more generally for tangled chains). Our main result is a combinatorial proof of the formula, which yields a simplification of the random sampler for tangled chains.