Counting rankings of tree-child networks

Rooted phylogenetic networks allow biologists to represent evolutionary relationships between present-day species by revealing ancestral speciation and hybridization events. A convenient and well-studied class of such networks are `tree-child networks' and a `ranking' of such a network is a temporal ordering of the ancestral speciation and hybridization events. In this short note, we investigate the question of counting such rankings on any given binary (or semi-binary) tree-child network. We also consider a class of binary tree-child networks that have exactly one ranking, and investigate further the relationship between ranked-tree child networks and the class of `normal' networks. Finally, we provide an explicit asymptotic expression for the expected number of rankings of a tree-child network chosen uniformly at random.