characterization of finite ultrametric spaces which are as rigid as possible also was obtained [@DPT2017FPTA] on the basis of the Gurvich—Vyalyi representation. Some other extremal properties of finite ultrametric spaces and related them properties of monotone rooted trees have been found in [@DP2020pNUAA]. The interconnections between the Gurvich—Vyalyi representation and the space of balls endowed with the Hausdorff metric are discussed in [@Dov2019pNUAA] (see also [@Qiu2009pNUAA; @Qiu2014pNUAA; @DP2018pNUAA; @Pet2018pNUAA; @Pet2013PoI]).

It is well-known that the sets of leaves of phylogenetic equidistant trees with additive metric are ultrametric. The finite equidistant trees can be considered as finite subtrees of the so-called $R$-trees (see [@Ber2019SMJ] for some interesting results related to $R$-trees and ultrametrics). The categorical equivalence of trees and ultrametric spaces was investigated in [@H04] and [@Lem2003AU].

It is interesting to note that, in fact, both the monotone trees and equidistant trees were used in the phylogenetic for the description of related ultrametric spaces long before the publication of paper [@GV2012DAM] (see, for example, [@SS2003OUP]). The monotone trees and the equidistant trees are dual in a certain sense, but it seems that the description of this duality can be found in Section 7.1 of book [@SS2003OUP] only.

In the present paper 