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

TL;DR

This work addresses the enumeration of seven subclasses of level-$2$ phylogenetic networks under planarity and structural constraints. It develops a generating-function framework, deriving an exponential generating function via a combinatorial decomposition and applying singular inversion to extract precise asymptotics of the counts, showing $N_n \sim c \, n^{n-1} \, \gamma^n$ with class-dependent constants. The results reveal how planarity constraints influence growth: terminal-planar reductions can notably lower growth rates for level-2 networks, while tree-child and galled constraints keep rates closer to level-1 benchmarks, highlighting the restrictive power of these topologies. The methodology and explicit asymptotics contribute broadly to the combinatorial understanding of constrained phylogenetic networks and can be adapted to other level-$k$ classes.

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: \[ N_n \sim c \cdot n^{n-1} \cdot γ^n, \] 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.

Figures (2)

• Figure 1: Examples of level-2 networks where reticulation nodes are represented by filled circles. (a) is a galled network which is not tree-child; (b) is a tree-child network which is not galled; (c) is a galled tree-child network. Both (b) and (c) are terminal and outer planar, whereas (a) is neither.
• Figure 2: Decomposition of the combinatorial structure for level-2 tree-child networks. Red edges indicate that they cannot be empty (must contain some insertion). In contrast, pairs of blue or green edges represent that, within each pair of edges of the same color, both cannot be empty simultaneously.

Theorems & Definitions (5)

• Proposition 1.1
• Theorem 2.1
• Remark 2.2
• Lemma 3.1
• Theorem 3.2