contrast with the weighted trees, the isomorphisms of labeled trees coincide with isometries of generated ultrametric spaces only for trees with one vertex.

In Section \[sec4\], after defining the concepts of equidistant weight and monotone labeling for the case of an arbitrary rooted tree, we find explicit formulas describing the transition between these weights and labelings in Proposition \[p6.1\]. Good functorial properties of such transition are described by Proposition \[p3.3\].

Theorem \[t6.4\], Theorem \[c3.13\] and Theorem \[t4.13\] are generalizations of the well-known fact about representation of finite ultrametric spaces by phylogenetic trees and by Gurvich—Vyalyi monotone trees.

Proposition \[p6.10\] describes the necessary and sufficient conditions under which isomorphism of equidistant trees (monotone trees) is equivalent to isometricity of corresponding ultrametric spaces.

Section \[sec5\] of the paper contains some characteristic properties of ultrametric spaces of leaves of equidistant rooted trees with pedant roots. In particular, Proposition \[p3.9\] describes a new characteristic property of stars in the language of equidistant weights.

Initial definitions and facts {#sec2}
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Let us recall some concepts from the theory of metric spaces and the graph theory.

A *metric* on a set $X$ is a function $d\colon X\times X\rightarrow \mathbb{R}^+$, $\mathbb R^+ = [0,\infty)$, such that for 