When k=o(√n), the number of binary phylogenetic networks with k reticulations on n taxa is asymptotic to binom(n,k) 2^{n+k-1/2} n^{n+k-1} e^{-n}.
Huson, Regula Rupp, and Celine Scornavacca.Phylogenetic Networks: Con- cepts, Algorithms and Applications
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Asymptotic Counting of Binary Phylogenetic Networks
When k=o(√n), the number of binary phylogenetic networks with k reticulations on n taxa is asymptotic to binom(n,k) 2^{n+k-1/2} n^{n+k-1} e^{-n}.