Asymptotic Enumeration of Subclasses of Level-2 Phylogenetic Networks

2026/01/29 by Hexuan Liu, Bing-Ze Lu, Taoyang Wu +1 · 1 voice
Mathematics · #math.CO

paper · pdf

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 ∼ c ⋅ n^n-1 ⋅ γⁿ, where c is a class-specific constant and γ 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.

Citations

• Galled Tree-Child Networks
• Planar Rooted Phylogenetic Networks
• Counting and enumerating tree-child networks and their subclasses
• Asymptotic Enumeration and Distributional Properties of Galled Networks
• On the Asymptotic Growth of the Number of Tree-Child Networks
• Counting and Enumerating Galled Networks
• Quarnet inference rules for level-1 networks
• Reconstructing phylogenetic level-1 networks from nondense binet and trinet sets
• Trinets encode tree-child and level-2 phylogenetic networks
• Constructing level-2 phylogenetic networks from triplets

Discussions

• Asymptotic Enumeration of Subclasses of Level-2 Phylogenetic Networks https://arxiv.org/abs/2601.21578v1 [bsky, 0 points, 0 comments]

Related

• Counting Phylogenetic Networks of level 1 and 2
• Counting Phylogenetic Networks with Few Reticulation Vertices: Tree-Child and Normal Networks
• Counting General Phylogenetic networks
• Counting Phylogenetic Networks with Few Reticulation Vertices: Exact Enumeration and Corrections
• Counting Phylogenetic Networks with Few Reticulation Vertices: A Second Approach