Order-Dependent Dissimilarity Measures on Phylogenetic Trees

TL;DR

This paper compares three vector-based dissimilarity measures for rooted phylogenetic trees with traditional metrics, showing their relationships and bounds when minimized over leaf orderings, and establishing equivalences with hybrid numbers.

Contribution

It demonstrates the relationships and bounds between vector-based measures and hybrid numbers for phylogenetic trees when optimized over leaf orderings.

Findings

Hybrid number equals HOP when minimized over all orderings.

Upper bounds relate vector measures to hybrid numbers.

Temporal tree-child hybrid number matches vector measures when minimized over specific orderings.

Abstract

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 equivalent to HOP, and 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 equivalent to each of the three vector-based measures.