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 2 γnlogn+O(n) , where the constant γ is 32 for general networks, and 54 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 2 2ℓlogℓ+O(ℓ) . Further we determine the typical numbers of leaves, tree vertices, and reticulation vertices for each of these classes of networks.