Distribution of external branch lengths in Yule trees

TL;DR

This paper analyzes the distribution of external branch lengths in Yule trees, revealing their asymptotic behavior and statistical properties, which are relevant for understanding genetic variation in population genetics models.

Contribution

It provides a detailed asymptotic distribution of external branch lengths in Yule trees, a novel analysis linking combinatorial tree features to population genetics.

Findings

External branch lengths follow a chi-distribution asymptotically.

Mean external branch length scales linearly with n.

Variance involves complex combinatorial terms.

Abstract

The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -- also called "histories" -- with a finite number $n$ of leaves. We study the lengths $\ell_1 > \ell_2 > ... > \ell_k > ...$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n \rightarrow \infty$, we show that the random variable $\ell_k$, once rescaled as $\frac{n-\ell_k}{\sqrt{n/2}}$, follows a $\chi$-distribution with $2k$ degrees of freedom, with mean $\mathbb E(\ell_k) \sim n$ and variance $\mathbb V(\ell_k) \sim n \big(k-\frac{\pi k^2}{16^k} \binom{2k}{k}^2\big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.