Counting phylogenetic networks

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 2γnlogn+O(n), where the constant γ is 3/2 for general networks, and 5/4 for tree-child and normal networks. We also show that the number of leaf-labelled tree-child and normal networks with ℓ leaves are both 22ℓlogℓ+O(ℓ). Further we determine the typical numbers of leaves, tree vertices, and reticulation vertices for each of these classes of networks.