Counting phylogenetic networks with few reticulation vertices: A second approach

Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for purpose modeling reticulate evolution. Recently, first author together with Gittenberger and Mansouri (2019) showed that number TCℓ,k tree-child networks ℓ leaves k reticulation vertices has first-order asymptotics TCℓ,k∼ck2eℓℓℓ+2k−1,(ℓ→∞).Moreover, they also computed ck k=1,2, 3. In this short note, we give a second approach to above result which is based on recent (algorithmic) counting due Cardona Zhang (2020). This capable giving simple, closed-form expression ck, namely, ck=2k−12/k! all k≥0.