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.
@article{10_37236_11438,
author = {Filippo Disanto and Michael Fuchs},
title = {Distribution of external branch lengths in {Yule} histories},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {3},
doi = {10.37236/11438},
zbl = {1534.92049},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11438/}
}
TY - JOUR
AU - Filippo Disanto
AU - Michael Fuchs
TI - Distribution of external branch lengths in Yule histories
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/11438/
DO - 10.37236/11438
ID - 10_37236_11438
ER -
%0 Journal Article
%A Filippo Disanto
%A Michael Fuchs
%T Distribution of external branch lengths in Yule histories
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/11438/
%R 10.37236/11438
%F 10_37236_11438
Filippo Disanto; Michael Fuchs. Distribution of external branch lengths in Yule histories. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11438
