[@PD2014JMS]).

If $X$ is a one-point set, then $T_X$ is the rooted tree consisting of the node $X$ with the label $\operatorname{diam}X=0$. Note that for the rooted trees consisting only of one node, we consider that this node is the root as well as a leaf.

Let $|X|\geqslant 2$. According to Theorem \[t13\] we have $G_{D,X} = G_{D,X}[X_1,...,X_k]$ and $k \geqslant 2$. In this case the root of the tree $T_X$ is labeled by $\operatorname{diam}X$ and $T_X$ has the nodes $X_1,...,X_k$ of the first level with the labels $$\label{e2.7}
l(X_i)= \operatorname{diam}X_i,$$ $i = 1,...,k$. The nodes of the first level with the label $0$ are leaves, and those indicated by strictly positive labels are internal nodes of the tree $T_X$. If the first level has no internal nodes, then the tree $T_X$ is constructed. Otherwise, by repeating the above-described procedure with $X_1,...,X_k$ instead of $X$, we obtain the nodes of the second level, etc. Since $X$ is finite, all vertices on some level will be leaves, and the construction of $T_X$ is completed.

The above-constructed labeled rooted tree $T_X$ is called the *representing tree* of the ultrametric space $(X, d)$.

In the next sections we will need several different concepts 