Labeled Trees Generating Separable and Locally Finite Ultrametrics

TL;DR

This paper explores how labeled trees can generate specific types of ultrametric spaces, providing characterizations and an analog of K"onig's Infinity Lemma for locally finite cases.

Contribution

It offers new characterizations of labeled trees that produce separable and locally finite ultrametric spaces, including an analog of K"onig's Infinity Lemma.

Findings

Characterization of labeled trees generating separable ultrametrics

Characterization of labeled trees generating locally finite ultrametrics

Establishment of an analog of K"onig's Infinity Lemma for locally finite ultrametrics

Abstract

We analyze the interplay between labeled trees and the ultrametric spaces they present. We provide characterizations of labeled trees that generate separable ultrametric spaces and those that generate locally finite ultrametric spaces. In particular, we establish an analog of K\"onig's Infinity Lemma for locally finite ultrametric spaces generated by labeled trees.