Locally finite ultrametric spaces and labeled trees

TL;DR

This paper characterizes when a locally finite ultrametric space can be generated by a labeled tree and constructs minimal such spaces containing a given finite ultrametric space as a subspace.

Contribution

It provides a necessary and sufficient condition for ultrametric spaces to be generated by labeled trees and constructs minimal extensions for finite ultrametric spaces.

Findings

Characterization of ultrametric spaces generated by labeled trees

Construction of minimal ultrametric extensions containing a given finite space

Uniqueness of minimal ultrametric extensions up to isometry

Abstract

It is shown that a locally finite ultrametric space $(X, d)$ is generated by labeled tree if and only if, for every open ball $B \subseteq X$, there is a point $c \in B$ such that $d(x, c) = \operatorname{diam} B$ whenever $x \in B$ and $x \neq c$. For every finite ultrametric space $Y$ we construct an ultrametric space $Z$ having the smallest possible number of points such that $Z$ is generated by labeled tree and $Y$ is isometric to a subspace of $Z$. It is proved that for a given $Y$, such a space $Z$ is unique up to isometry.