all $x$, $y$, $z \in X$

1.  $d(x,y)=d(y,x)$,

2.  $(d(x,y)=0)\Leftrightarrow (x=y)$,

3.  $d(x,y)\leq d(x,z) + d(z,y)$.

The quantity $$\operatorname{diam}X=\sup\{d(x,y)\colon x,y\in X\}$$ is the *diameter* of $(X,d)$.

A metric space $(X, d)$ is *ultrametric* if the *strong triangle inequality* $$d(x,y)\leq \max \{d(x,z),d(z,y)\}.$$ holds for all $x$, $y$, $z \in X$. In this case $d$ is called *an ultrametric* on $X$ and $(X, d)$ is an *ultrametric space*.

\[d1.1\] Metric spaces $(X, d)$ and $(Y, \rho)$ are *isometric* if there is a bijective mapping $\Phi \colon X \to Y$ such that $$d(x,y) = \rho(\Phi(x), \Phi(y))$$ holds for all $x$, $y \in X$. In this case we write $(X, d) \simeq (Y, \rho)$ and say that $\Phi$ is an *isometry* of $(X, d)$ and $(Y, \rho)$.

A *graph* is a pair $(V,E)$ consisting of a nonempty set $V$ and a (probably empty) set $E$ whose elements are unordered pairs of different points from $V$. For a graph $G=(V,E)$, the sets $V=V(G)$ and $E=E(G)$ are called *the set of vertices (or nodes)* and *the set of edges*, respectively. We say that $G$ is *nonempty* if $E(G) \neq \varnothing$. If $\{x,y\} \in E(G)$, then the vertices $x$ and $y$ are *adjacent*. Recall that a *path* 