Finite Ultrametric Balls

TL;DR

This paper characterizes when a family of subsets of a finite set corresponds exactly to the family of all ultrametric balls and describes how to construct the representing tree of the associated ultrametric space.

Contribution

It provides necessary and sufficient conditions for a family of subsets to be the family of ultrametric balls and details the construction of the representing tree for the Hausdorff distance space.

Findings

Characterization of families of subsets as ultrametric balls

Method to construct the representing tree from the original ultrametric space

Description of the relationship between the representing trees of the space and its Hausdorff distance space

Abstract

The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family $\mathbf{B}_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the representing tree of the ultrametric space $(\mathbf{B}_{X}, d_H)$ with the Hausdorff distance $d_H$ can be obtained from the representing tree $T_X$ of ultrametric space $(X, d)$ by adding a leaf to every internal vertex of $T_X$.