Distribution of external branch lengths in Yule trees

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 $n\to \infty$, 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 $\mathbb{V}({\ell }_k)\sim n(k-\frac{\pi k^2}{16^k}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2)$. 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.