Exact and Asymptotic Counts of Binary Phylogenetic Networks with a Few Reticulation Events

Phylogenetic networks are powerful models for representing evolutionary history in studies of genome evolution involving reticulation events. The complex structure of phylogenetic networks makes their inference challenging, especially for large numbers of taxa. To better understand the full space of phylogenetic networks, we derive closed-form formulas for counting binary phylogenetic networks with two and three reticulation events. In addition, we also use our approach to give a simple proof of the known asymptotic result for a fixed number of reticulations.