The Space of Equidistant Phylogenetic Cactuses.

An - is a type of rooted, arc-weighted, directed acyclic graph with leaf set , that is used in biology to represent the evolutionary history of a set  of species. In this paper, we introduce and investigate the space of equidistant -cactuses. This space contains, as a subset, the space of ultrametric trees on that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning rooted -cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of  that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted -trees in terms of partitions of , which provides an alternative proof that the space of ultrametric trees on is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant -cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.