---
abstract: 'We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated by labelings given on the vertices of trees. The obtained results generalized some facts previously known for phylogenetic trees and for Gurvich—Vyalyi monotone trees.'
address: 'Functions Theory Department, Institute of Applied Mathematics and Mechanics of NASU, Dobrovolskogo str. 1, Slovyansk 84100, Ukraine'
author:
- Oleksiy Dovgoshey
title: Isomorphism of trees and isometry of ultrametric spaces
---

Introduction
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In 2001 at the Workshop on General Algebra the attention of experts on the theory of lattices was paid to the following problem of I. M. Gelfand: Using graph theory describe up to isometry all finite ultrametric spaces [@Lem2001]. An appropriate representation of ultrametric spaces by monotone rooted trees was proposed by V. Gurvich and M. Vyalyi in [@GV2012DAM]. A simple geometric description of Gurvich—Vyalyi representing trees was found in [@PD2014JMS]. This description allows us effectively use the Gurvich—Vyalyi representation in various problems associated with finite ultrametric spaces. In particular, this leads to a graph-theoretic interpretation of the Gomory-Hu Inequality [@DPT2015]. A 