Counting Phylogenetic Networks with Few Reticulation Vertices: A Second Approach

TL;DR

This paper presents a new, simpler method for counting tree-child phylogenetic networks with a given number of reticulation vertices, providing explicit formulas for their asymptotic enumeration.

Contribution

It introduces a second approach based on recent algorithms that yields a closed-form expression for the asymptotic constants in counting such networks.

Findings

Derived a simple closed-form expression for the asymptotic constants c_k.

Validated the second approach as effective for counting tree-child networks.

Provided insights into the enumeration of phylogenetic networks with reticulation vertices.

Abstract

Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number ${\rm TC}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices has the first-order asymptotics \[ {\rm TC}_{\ell,k}\sim c_k\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad (\ell\rightarrow\infty). \] Moreover, they also computed $c_k$ for $k=1,2,$ and $3$. In this short note, we give a second approach to the above result which is based on a recent (algorithmic) approach for the counting of tree-child networks due to Cardona and Zhang (2020). This second approach is also capable of giving a simple, closed-form expression for $c_k$, namely, $c_k=2^{k-1}\sqrt{2}/k!$ for all $k\geq 0$.