Counting Phylogenetic Networks of level 1 and 2

TL;DR

This paper develops enumeration formulas and analyzes the distribution of parameters for level-1 and level-2 phylogenetic networks, advancing understanding of their combinatorial properties.

Contribution

It provides exact and asymptotic enumeration formulas for these networks and proves normal distribution of certain parameters, using combinatorial and analytic methods.

Findings

Exact enumeration formulas for level-1 and level-2 networks

Asymptotic formulas and normal distribution of network parameters

Recursive combinatorial descriptions of the networks

Abstract

Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can all be modeled as families of graphs with labeled leaves. In this paper, we focus on rooted and unrooted level-k networks and provide enumeration formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2 phylogenetic networks with a given number of leaves. We also prove that the distribution of some parameters of these networks (such as their number of cycles) are asymptotically normally distributed. These results are obtained by first providing a recursive description (also called combinatorial specification) of our networks, and by next applying classical methods of enumerative, symbolic and analytic combinatorics.