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arXiv · 2508.13414

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

Charles Semple·Kristina Wicke
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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.

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Charles Semple, Kristina Wicke. 2025-08-19. A sharp lower bound for the number of phylogenetic trees displayed by a tree-child network. https://arxiv.org/abs/2508.13414

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