Can Multiple Phylogenetic Trees Be Displayed in a Tree-Child Network Simultaneously?

TL;DR

This paper investigates conditions under which multiple phylogenetic trees can be simultaneously represented in a single tree-child network, proving that any two trees can always be displayed and extending results to non-binary networks.

Contribution

It establishes necessary conditions for displaying multiple phylogenetic trees in tree-child networks and proves that any two trees can always be simultaneously displayed.

Findings

Any two phylogenetic trees can be displayed in some tree-child network.

Any set of multiple binary trees can be displayed in a non-binary tree-child network.

Necessary conditions for multiple trees being simultaneously displayed are provided.

Abstract

A binary phylogenetic network on a taxon set $X$ is a rooted acyclic digraph in which the degree of each nonleaf node is three and its leaves (i.e.degree-one nodes) are uniquely labeled with the taxa of $X$. It is tree-child if each nonleaf node has at least one child of indegree one. A set of binary phylogenetic trees may or may not be simultaneously displayed in a binary tree-child network. Necessary conditions for multiple phylogenetic trees being simultaneously displayed in a tree-child network are given here. In particular, it is proved that any two phylogenetic trees can always simultaneously be displayed in some tree-child network on the same taxa set. It is also proved that any set of multiple binary phylogenetic trees can always simultaneously be displayed in some non-binary tree-child network on the same taxa set, where each nonleaf node is of either indegree one and outdegree two or indegree at least two and outdegree out.