Order-Dependent dissimilarity measures on phylogenetic trees

Ordered leaf attachment, Phylo2Vec, and HOP are three recently introduced vector representations for rooted phylogenetic trees where the representation is determined by an ordering of the underlying leaf set X. Comparing the vectors of two rooted phylogenetic X-trees T and T′ for a fixed ordering on X leads to polynomial-time computable measure for the dissimilarity of T and T′, albeit dependent on the choice of the leaf ordering. For each of ordered leaf attachment, Phylo2Vec, and HOP, we compare this measure with the rooted subtree prune and regraft distance (rSPR), the hybrid number, and the temporal tree-child hybrid number of T and T′. Although there is no direct relationship between rSPR and any of the three vector-based measures, we show that, when minimized over all orderings, the hybrid number is equal to HOP and is an upper bound on the other two. Moreover, when minimized over all orderings induced by common cherry-picking sequences of T and T′, the temporal tree-child hybrid number of T and T′ is equal to each of the three vector-based measures.