Counting phylogenetic networks with few reticulation vertices: A second approach

Abstract

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