Labeled trees generating complete, compact, and discrete ultrametric spaces

TL;DR

This paper explores how labeled trees can generate various types of ultrametric spaces, providing characterizations and properties that link tree structures to metric space features.

Contribution

It offers a comprehensive characterization of labeled trees that generate complete, compact, and discrete ultrametric spaces, advancing the understanding of their interrelations.

Findings

Labeled trees generating complete ultrametrics are characterized.

Labeled trees generating compact ultrametrics are characterized.

Every ultrametric space from labeled trees contains a dense discrete subspace.

Abstract

We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The labeled trees, which generate complete ultrametrics, totally bounded ultrametrics, and discrete ones, are characterized up to isomorphism. As corollary, we obtain a characterization of labeled trees generating compact ultrametrics, and discrete totally bounded ultrametrics. It is also shown that every ultrametric space generated by labeled tree contains a dense discrete subspace.