Counting Tree-Child Networks and Their Subclasses

TL;DR

This paper develops enumeration formulas for counting various classes of phylogenetic networks, including tree-child and galled trees, by analyzing their structural properties and component graphs.

Contribution

It introduces explicit counting formulas for tree-child and galled trees, expanding the understanding of their combinatorial structures.

Findings

Derived enumeration formulas for tree-child networks.

Established relationships between galled trees and ordered trees.

Provided counts for networks with limited reticulations.

Abstract

Galled trees are studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into tree-child, galled and reticulation-visible network classes by relaxing a structural condition imposed on galled trees. We count tree-child networks through enumerating their component graphs. Explicit counting formulas are also given for galled trees through their relationship to ordered trees, phylogenetic networks with few reticulations and phylogenetic networks in which the child of each reticulation is a leaf.