Extremal properties and morphisms of finite ultrametric spaces and their representing trees

TL;DR

This paper investigates extremal properties of finite ultrametric spaces and their representing trees, introduces weak similarity, and explores conditions linking tree isomorphisms to ultrametric isometries.

Contribution

It introduces the notion of weak similarity for finite ultrametric spaces and establishes conditions under which representing tree isomorphisms imply ultrametric isometries.

Findings

Finite rooted trees are isomorphic to trees of nonsingular balls in ultrametric spaces.

Conditions are identified where tree isomorphism implies ultrametric isometry.

Morphisms of labeled rooted trees relate to weak similarities of ultrametric spaces.

Abstract

We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.