A Short Note on the Exact Counting of Tree-Child Networks

TL;DR

This paper proves a conjecture relating tree-child networks to specific words, focusing on the class of one-component networks, which are significant in phylogenetics for modeling complex evolutionary histories.

Contribution

It establishes the conjecture for one-component tree-child networks, advancing understanding of their combinatorial structure in phylogenetics.

Findings

Proved the conjecture for one-component tree-child networks.

Enhanced understanding of the combinatorial properties of these networks.

Contributed to the mathematical foundation of phylogenetic network modeling.

Abstract

Tree-child networks are an important network class which are used in phylogenetics to model reticulate evolution. In a recent paper, Pons and Batle (2021) conjectured a relation between tree-child networks and certain words. In this short note, we prove their conjecture for the (important) class of one-component tree-child networks.