Counting Phylogenetic Networks

Abstract

We give approximate counting formulae for the numbers of labelled general, treechild, and normal (binary) phylogenetic networks on n vertices. These formulae are of the form $${2^{\gamma n {\rm log}n+O(n)}}$$ , where the constant $${\gamma}$$ is $${\frac{3}{2}}$$ for general networks, and $${\frac{5}{4}}$$ for tree-child and normal networks. We also show that the number of leaf-labelled tree-child and normal networks with $${\ell}$$ leaves are both $${{2}^{2 \ell {\rm log} \ell +O( \ell )}}$$ . Further we determine the typical numbers of leaves, tree vertices, and reticulation vertices for each of these classes of networks.