Asymptotic enumeration of normal and hybridization networks via tree decoration

Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place $k$ arcs between the edges of a rooted binary phylogenetic tree with $n$ leaves. The resulting directed graph may fail to be a phylogenetic network, and even when it is (and thereby a `tree-based' network), it may fail to be a tree-child or normal network. In this paper, we first show that if $k$ is fixed, the proportion of arc placements that result in a normal network tends to 1 as $n$ grows. From this result, the asymptotic enumeration of normal networks becomes straightforward and provides a transparent meaning to the combinatorial terms that arise. Moreover, the approach extends to allow $k$ to grow with $n$ (at the rate $o(n^\frac{1}{3})$), which was not handled in earlier work. We also investigate a subclass of normal networks of particular relevance in biology (hybridization networks) and establish that the same asymptotic results apply.