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arXiv · 2601.21578

Asymptotic Enumeration of Subclasses of Level-$2$ Phylogenetic Networks

Hexuan Liu·Bing-Ze Lu·Taoyang Wu·Guan-Ru Yu
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Abstract

This paper studies the enumeration of seven subclasses of level-$2$ phylogenetic networks under various planarity and structural constraints, including terminal planar, tree-child, and galled networks. We derive their exponential generating functions, recurrence relations, and asymptotic formulas. Specifically, we show that the number of networks of size $n$ in each class follows: \[ N_n \sim c \cdot n^{n-1} \cdot \gamma^n, \] where $c$ is a class-specific constant and $\gamma$ is the corresponding growth rate. Our results reveal that being terminal planar can significantly reduce the growth rate of general level-2 networks, but has only a minor effect on the growth rates of tree-child and galled level-2 networks. Notably, the growth rate of 3.83 for level-$2$ terminal planar galled tree-child networks is remarkably close to the rate of 2.94 for level-$1$ networks.

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Hexuan Liu, Bing-Ze Lu, Taoyang Wu, Guan-Ru Yu. 2026-01-29. Asymptotic Enumeration of Subclasses of Level-$2$ Phylogenetic Networks. https://arxiv.org/abs/2601.21578

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