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

TL;DR

This paper provides asymptotic enumeration formulas for seven subclasses of level-2 phylogenetic networks, revealing how structural constraints influence their growth rates and offering insights into their combinatorial complexity.

Contribution

It derives explicit asymptotic formulas, recurrence relations, and generating functions for subclasses of level-2 phylogenetic networks, highlighting the impact of planarity and structural constraints.

Findings

Terminal planar networks have significantly reduced growth rates.

Tree-child and galled networks are less affected by planarity constraints.

Growth rate of level-2 terminal planar galled tree-child networks is close to level-1 networks.

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.