Properties and Morphisms of Finite Ultrametric Spaces and Their Representing Trees

Abstract

We study extremal properties of finite ultrametric spaces X and related properties of representing trees T X . The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees T X and T Y implies the isometricity of ultrametric spaces X and Y .2010 Mathematics Subject Classification. 54E35, 05C05. Key words and phrases. Finite ultrametric space, sharp inequality for finite ultrametric space, isometry, ball-preserving mapping, isomorphism of rooted trees.

Finite Ultrametric Balls

“…The following theorem and their corollary can be found in [16] but in view of the importance of this result in the context of the paper, we present it with a short proof. and, in addition, the inequality…”

“…Following [40] we will say that these trees are representing trees of spaces (X, d). The present paper can be considered as a development of studies initiated at [22] and continued at [10,16,18,[39][40][41].…”

“…[16]). Let T = T (r, l T ) be a finite labeled rooted tree with the root r and the labeling l T : V (T ) → R + .…”

From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

“…The following theorem is a simple modification of Theorem 2.7 [10]. and, in addition, the inequality l(v) < l(u) holds whenever v is a direct successor of u.…”

Isomorphism of Trees and Isometry of Ultrametric Spaces

“…Let T = T (r) be a rooted tree, let v ∈ V (T ) and let L(T ) be the set of all leaves of T . As in [7,[9][10][11][12][13][14], we will denote by δ…”