Enumeration of $d$-combining Tree-Child Networks

TL;DR

This paper extends the enumeration of tree-child networks to include $d$-combining cases where reticulation nodes have $d$ parents, and explores the distribution of reticulation nodes in random networks.

Contribution

It introduces the enumeration of $d$-combining tree-child networks and investigates their properties, expanding prior work limited to bicombining networks.

Findings

Enumeration formulas for $d$-combining tree-child networks.

Conjectures on the distribution of reticulation nodes in random networks.

Insights into the structural complexity of these networks.

Abstract

Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for {\it bicombining tree-child networks} which are tree-child networks with every reticulation node having exactly two parents. In this paper, we extend these studies to {\it $d$-combining tree-child networks} where every reticulation node has now $d\geq 2$ parents. Moreover, we also give results and conjectures on the distributional behavior of the number of reticulation nodes of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves.