Asymptotic enumeration and distributional properties of galled networks

Abstract

We show a first-order asymptotics result for the number of galled networks with n leaves. This is the first class of phylogenetic networks of large size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of galled networks with n leaves chosen uniformly at random. These results are obtained by performing an asymptotic analysis of a recent approach of Gunawan, Rathin, and Zhang (2020) [12] which was devised for the purpose of (exactly) counting galled networks. Moreover, an old result of Bender and Richmond (1984) [1] plays a crucial role in our proofs, too.