Counting rankings of tree-child networks
Qiang Zhang, Mike Steel
TL;DR
The paper develops a framework to count temporal rankings of tree-child phylogenetic networks by relating rankings to topological orderings of an auxiliary graph \boldsymbol{\Psi}(N). It proves that every rankable separated tree-child network is normal and shows a bijection between rankings and orderings when \boldsymbol{\Psi}(N) is a rooted tree, enabling explicit counting formulas. For binary tree-child networks with $n$ leaves and $k$ reticulations, it derives an exact asymptotic formula for the expected number of rankings as $n\to\infty$ with fixed $k$, namely $\mathbb{E}[X_{n,k}]\sim \frac{1}{4^k}\cdot\frac{n!}{\binom{2n-2}{n-1}}$, and shows that the probability of having at least one ranking tends to 1. The results combine combinatorial enumeration with known asymptotics for Stirling numbers and rooted binary trees, and suggest future work on the computational complexity of ranking counts and the distribution of rankings in large networks.
Abstract
Rooted phylogenetic networks allow biologists to represent evolutionary relationships between present-day species by revealing ancestral speciation and hybridization events. A convenient and well-studied class of such networks are `tree-child networks' and a `ranking' of such a network is a temporal ordering of the ancestral speciation and hybridization events. In this short note, we investigate the question of counting such rankings on any given binary (or semi-binary) tree-child network. We also consider a class of binary tree-child networks that have exactly one ranking, and investigate further the relationship between ranked-tree child networks and the class of `normal' networks. Finally, we provide an explicit asymptotic expression for the expected number of rankings of a tree-child network chosen uniformly at random.