Recovering Symbolically Dated, Rooted Trees from Symbolic Ultrametrics

Abstract

A well known result from cluster theory states that there is a 1-to-1 correspondence between dated, compact, rooted trees and ultrametrics. In this paper, we generalize this result yielding a canonical 1-to-1 correspondence between symbolically dated trees and symbolic ultrametrics, using an arbitrary set as the set of (possible) dates or values. It turns out that a rather unexpected new condition is needed to properly define symbolic ultrametrics so that the above correspondence holds. In the second part of the paper, we use our main result to derive, as a corollary, a theorem by H. J. Bandelt and M. A. Steel regarding a canonical 1-to-1 correspondence between additive trees and metrics satisfying the 4-point condition, both taking their values in abelian monoids. © 1998 Academic Press.