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

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.

💡 Research Summary

The paper tackles the notoriously difficult problem of counting phylogenetic networks by focusing on a well‑structured subclass: level‑2 networks. A level‑k network is defined by the restriction that every biconnected component contains at most k reticulation (hybridisation) nodes. While level‑1 networks coincide with galled trees and are relatively easy to enumerate, level‑2 networks already exhibit a rich combinatorial variety. The authors further refine the landscape by imposing planarity constraints (general planar, upward planar, terminal planar, outer planar) and structural constraints (tree‑child, galled, and their intersection, called GTC – galled tree‑child networks). In total, seven distinct subclasses are studied.

Methodology.
The core of the analysis is the construction of exponential generating functions (EGFs) for each subclass. The authors decompose a network into elementary building blocks – leaves, tree nodes (one parent, two children) and reticulation nodes (two parents, one child) – and encode admissible ways of gluing these blocks together. Figure 2 illustrates the decomposition for level‑2 tree‑child networks; coloured edges indicate dependencies that prevent certain substructures from being empty simultaneously. Translating the combinatorial decomposition into a functional equation yields a highly non‑linear relation such as

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