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Phylogenetic networks are a special type of graph that generalizephylogenetic trees and that are used to model non-treelike evolutionaryprocesses such as recombination and hybridization. In this paper, we considerunrooted phylogenetic networks, i.e. simple, connected graphs$\mathcal{N}=(V,E)$ with leaf set $X$, for $X$ some set of species, in whichevery internal vertex in $\mathcal{N}$ has degree three. One approach used toconstruct such phylogenetic networks is to take as input a collection$\mathcal{P}$ of phylogenetic trees and to look for a network $\mathcal{N}$that contains each tree in $\mathcal{P}$ and that minimizes the quantity$r(\mathcal{N}) = |E|-(|V|-1)$ over all such networks. Such a network alwaysexists, and the quantity $r(\mathcal{N})$ for an optimal network $\mathcal{N}$is called the hybrid number of $\mathcal{P}$. In this paper, we give a newcharacterization for the hybrid number in case $\mathcal{P}$ consists of twotrees. This characterization is given in terms of a cherry picking sequence forthe two trees, although to prove that our characterization holds we need todefine the sequence more generally for two forests. Cherry picking sequenceshave been intensively studied for collections of rooted phylogenetic trees, butour new sequences are the first variant of this concept that can be applied inthe unrooted setting. Since the hybrid number of two trees is equal to thewell-known tree bisection and reconnect distance between the two trees, our newcharacterization also provides an alternative way to understand this importanttree distance.

Cherry picking in forests: A new characterization for the unrooted hybrid number of two phylogenetic trees