The space of equidistant phylogenetic cactuses

Katharina T Huber, Vincent Moulton, Megan Owen, Andreas Spillner, Katherine St. John

Research output: Contribution to journalArticlepeer-review


We introduce and investigate the space of equidistant X-cactuses. These are rooted, arc weighted, phylogenetic networks with leaf set X, where X is a finite set of species, and all leaves have the same distance from the root. The space contains as a subset the space of ultrametric trees on X 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 ranked rooted X-cactuses. In particular, we show that such networks can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of X that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted X-trees in terms of partitions of X, which provides an alternative proof that the space of ultrametric trees on X is CAT(0). As with spaces of phylogenetic trees, we expect that our results should provide the basis for and new directions in performing statistical analyses for collections of phylogenetic networks with arc lengths.
Original languageEnglish
JournalAnnals of Combinatorics
Publication statusAccepted/In press - 24 May 2023


  • phylogenetic network
  • network space
  • combinatorial encoding
  • CAT(0)-metric space

Cite this