Ball-preserving mappings of finite ultrametric spaces

TL;DR

This paper establishes a characterization of finite ultrametric spaces through the isomorphism of their representing rooted trees, using the concept of ball-preserving bijections.

Contribution

It introduces a necessary and sufficient condition for the isomorphism of finite ultrametric spaces based on ball-preserving bijections.

Findings

Rooted trees $T_X$ and $T_Y$ are isomorphic iff a ball-preserving bijection exists.

Provides a new criterion for ultrametric space isomorphism.

Connects ultrametric space structure with rooted tree isomorphism.

Abstract

It is shown that the rooted trees $T_X$ and $T_Y$ representing finite ultrametric spaces $X$ and $Y$ are isomorphic if and only if there exists a ball-preserving bijection $F:X\to Y$.