Counting doubly labelled trees and one-component networks

A phylogenetic network is one-component if the child of every reticulation is a leaf. This class of networks has been widely used as building blocks in the combinatorial study of phylogenetic networks. We establish a bijection between one-component networks on k taxa and rooted binary multi-labelled trees in which every label occurs exactly twice. Using this bijection and Burnside's lemma, we derive a closed formula and a two-term recurrence for the number of one-component networks. We further obtain explicit formulas for counting one-component networks with a prescribed number of reticulations.