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An equidistant X-cactus is a type of rooted, arc-weighted, directed acyclic graph with leaf set X, that is used in biology to represent the evolutionary history of a set X of species. In this paper, we introduce and investigate the space of equidistant X-cactuses. This 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 graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of sub- sets 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). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant X-cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks. 

Abstract Popular optimality criteria for phylogenetic trees focus on sequences of characters that are applicable to all the taxa. As studies grow in breadth, it can be the case that some characters are applicable for a portion of the taxa and inapplicable for others. Past work has explored the limitations of treating inapplicable characters as missing data, noting that this strategy may favor trees where internal nodes are assigned impossible states, where the arrangement of taxa within subclades is unduly influenced by variation in distant parts of the tree, and/or where taxa that otherwise share most primary characters are grouped distantly. Approaches that avoid the first two problems have recently been proposed. Here, we propose an alternative approach which avoids all three problems. We focus on data matrices that use reductive coding of traits, that is, explicitly incorporate the innate hierarchy induced by inapplicability, and as such our approach extend to hierarchical characters, in general. In the spirit of maximum parsimony, the proposed criterion seeks the phylogenetic tree with the minimal changes across any tree branch, but where changes are defined in terms of dissimilarity metrics that weigh the effects of inapplicable characters. The approach can accommodate binary, multistate, ordered, unordered, and polymorphic characters. We give a polynomial-time algorithm, inspired by Fitch’s algorithm, to score trees under a family of dissimilarity metrics, and prove its correctness. We show that the resulting optimality criteria is computationally hard, by reduction to the NP-hardness of the maximum parsimony optimality criteria. We demonstrate our approach using synthetic and empirical data sets and compare the results with other recently proposed methods for choosing optimal phylogenetic trees when the data includes hierarchical characters. [Character optimization, dissimilarity metrics, hierarchical characters, inapplicable data, phylogenetic tree search.]