Distribution of External Branch Lengths in Yule Histories

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 $\ell_1 > \ell_2 > \cdots > \ell_k > \cdots$ 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.