Counting Tree-Child Networks and Their Subclasses

Galled trees are studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into tree-child, galled and reticulation-visible network classes by relaxing a structural condition imposed on galled trees. We count tree-child networks through enumerating their component graphs. Explicit counting formulas are also given for galled trees through their relationship to ordered trees, phylogenetic networks with few reticulations and phylogenetic networks in which the child of each reticulation is a leaf.