we discuss the interrelations between the weighted trees with additive metrics and labeled trees with corresponding ultrametrics. In particular, the duality of equidistant trees and monotone trees is studied in details for trees which are more general than the classical phylogenetic trees.

The paper is organized as follows.

Section \[sec2\] contains some standard definitions from the theory of metric spaces and the theory of graphs. A short list of known properties of Gurvich—Vyalyi representing trees is also given there.

Section \[sec3\] deals with the weights and labelings on unrooted trees. In Theorem \[t3.4\] we prove that, for all weights, the isomorphisms of weighted trees coincide with isometries of metric spaces endowed with the corresponding additive metrics and, in Proposition \[p3.4\], we show that this property characterizes the trees among all connected weighted graphs. Proposition \[p3.5\] describes conditions under which the labelings on the vertex sets of trees generate ultrametrics. In Theorem \[t3.9\] it is shown that under the same conditions every connected labeled graph $G$ contains a spanning tree $T$ such that labeling, induced on $T$, generates the same ultrametric as an original labeling on $G$. Theorem \[t3.12\], one of the main results of the section, shows that, in the 