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.
Highlights
In 2001 Gelfand caught the attention of the experts in the theory of lattices with the following problem: using graph theory describe up to isometry all finite ultrametric spaces [18]
A simple geometric description of Gurvich—Vyalyi representing trees was found in [21]. This description allows us to use effectively the Gurvich— Vyalyi representation in various problems associated with finite ultrametric spaces
A characterization of finite ultrametric spaces which are as rigid as possible was obtained [14] on the basis of the Gurvich—Vyalyi representation
Summary
In 2001 Gelfand caught the attention of the experts in the theory of lattices with the following problem: using graph theory describe up to isometry all finite ultrametric spaces [18]. The interconnections between the Gurvich—Vyalyi representation and the space of balls endowed with the Hausdorff metric are discussed in [7] (see [10, 20, 22,23,24]) It is well-known that the sets of leaves of phylogenetic equidistant trees with shortest-path metric are ultrametric. Theorem 3.7, one of the main results of the section, shows that, in the contrast with the weighted trees, the isomorphisms of labeled trees coincide with isometries of generated ultrametric spaces only for trees with one vertex. Theorem 4.3, Theorem 4.4 and Theorem 4.5 are generalizations of the well-known fact about representation of finite ultrametric spaces by phylogenetic trees and by Gurvich— Vyalyi monotone trees. Proposition 4.5 describes the necessary and sufficient conditions under which isomorphism of equidistant trees (monotone trees) is equivalent to isometricity of corresponding ultrametric spaces. (X, d) (Y, ρ) — the metric spaces (X, d) and (Y, ρ) are isometric, 3
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