A sharp lower bound for the number of phylogenetic trees displayed by a tree-child network

TL;DR

This paper establishes new lower bounds on the number of phylogenetic trees displayed by tree-child networks with no underlying 3-cycles, providing sharp bounds and characterizations.

Contribution

It introduces the first sharp lower bounds for the number of displayed trees in tree-child networks without 3-cycles, extending known results for normal networks.

Findings

Lower bound of 2^{k/2} trees for even k

Lower bound of (3/2√2)2^{k/2} trees for odd k

Characterization of networks attaining these bounds

Abstract

A normal (phylogenetic) network with $k$ reticulations displays $2^k$ phylogenetic trees. In this paper, we establish an analogous result for tree-child (phylogenetic) networks with no underlying $3$-cycles. In particular, we show that a tree-child network with $k\ge 2$ reticulations and no underlying $3$-cycles displays at least $2^{k/2}$ phylogenetic trees if $k$ is even and at least $\frac{3}{2\sqrt{2}}2^{k/2}$ if $k$ is odd. Moreover, we show that these bounds are sharp and characterise the tree-child networks that attain these bounds.