(unrooted, without labelings) trees.

2.  \[p2.13:s2\] There is an ultrametric space $(X, d)$ such that the diametrical graph $G_{D, X}$ is complete $k$-partite with $k \geqslant 3$ and $T_X$ and $T$ are isomorphic as free (unrooted, without labelings) trees.

Let $(X,d)$ be an ultrametric space. Recall that a *ball* with a radius $r \geqslant 0$ and a center $c\in X$ is the set $$B_r(c)=\{x\in X\colon d(x,c)\leqslant r\}.$$ The *ballean* $\mathbf{B}_X$ of the ultrametric space $(X,d)$ is the set of all balls of $(X,d)$. Every one-point subset of $X$ belongs to $\mathbf{B}_X$, this is a *singular* ball in $X$.

Let $T = T(r)$ be a rooted tree. For every vertex $v$ of $T$ we denote by $T_v$ the subtree of $T$ such that $v$ is the root of $T_v$, $$\label{e2.5}
V(T_v) = \{u \in V(T) \colon u = v \text{ or \(u\) is a successor of } v\},$$ and satisfying $$\label{e2.6}
(\{v_1, v_2\} \in E(T_v)) \Leftrightarrow (\{v_1, v_2\} \in E(T(r)))$$ for all $v_1$, $v_2 \in V(T_v)$. In this situation Charles Semple and Mike Steel say that $T_v$ is a rooted subtree of $T(r)$ lying below $v$ [@SS2003OUP p. 9]. See Figure \[fig1\] for an example of such a rooted subtree.

In 