Note on the Maximum Number of Trees Displayed by a Tree-Child Network

2026-06-12

AI Summary

This study investigates the maximum number of distinct rooted binary phylogenetic trees that can be displayed by a binary phylogenetic network with \(n\) leaves. By integrating combinatorial arguments with phylogenetic network theory and employing structural replacements based on cherries and reticulated cherries, the authors establish that this maximum equals \(2^{n-1} - 1\), and demonstrate that the bound is tight. They further characterize the structural properties of networks achieving this upper bound and introduce a canonical algorithm to identify networks that display a unique tree. This work provides the first exact upper bound on the tree-displaying capacity of tree-child networks together with a precise structural characterization of the extremal cases.

Abstract

In this note, we show that, for all $n\ge 2$, the number of distinct rooted binary phylogenetic $X$-trees displayed by a binary tree-child network $\mathcal{N}$ on $X$ with $n$ leaves is at most $2^{n-1}-1$ and that this upper bound is sharp. Furthermore, if $\mathcal{N}$ displays exactly $2^{n-1}-1$ such trees, then exactly one rooted binary phylogenetic $X$-tree is displayed twice, and this tree can be canonically found by iteratively replacing a reticulated cherry with a cherry.

Problem

• tree-child network
• phylogenetic tree
• displayed trees
• reticulated cherry
• upper bound

Innovation

• tree-child network
• phylogenetic tree
• reticulated cherry
• upper bound
• binary network