# Counting Rankings of Tree-Child Networks

## TL;DR

This paper explores how to count the number of ways to order evolutionary events in a specific type of phylogenetic network called a tree-child network.

## Contribution

The paper introduces a method to count rankings of tree-child networks and derives an asymptotic expression for the expected number of rankings.

## Key findings

- An explicit asymptotic formula for the expected number of rankings of a random tree-child network is derived.
- The relationship between rankable tree-child networks and normal networks is investigated.
- A method for counting rankings on binary or semi-binary tree-child networks is presented.

## 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 investigate the relationship between rankable 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.