Arboreal Ultrametrics

TL;DR

This paper introduces arboreal ultrametrics, a generalization of ultrametrics represented by rooted networks with multiple roots, and characterizes their properties and unique root insertions in phylogenetic trees.

Contribution

It defines arboreal ultrametrics, characterizes when partial distances are arboreal ultrametrics, and proves the uniqueness of root insertions in phylogenetic trees.

Findings

Characterization of arboreal ultrametrics from partial distances.

Existence and uniqueness of root insertions in phylogenetic trees.

Application of arboreal ultrametrics in phylogenetics and graph theory.

Abstract

Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.