\rho_{w_2})$ is an isomorphism of $G_1(w_1)$ and $G_2(w_2)$.

In the rest of the section we consider the labeled, unrooted trees and related them ultrametric spaces and find some modifications of Theorem \[t2.9\], Theorem \[t3.4\], Proposition \[p3.4\] and Problem \[pr1\] in this case.

Let $T = T(l)$ be a labeled tree with the labeling $l \colon E(T) \to \mathbb{R}^{+}$ and let $d_l \colon V(T) \times V(T) \to \mathbb{R}^{+}$ be a mapping defined as $$\label{e3.11}
d_l(u, v) = \begin{cases}
0, & \text{if } u = v\\
\max\limits_{v^{*} \in V(P)} l(v^{*}), & \text{if } u \neq v,
\end{cases}$$ where $P$ is the unique path joining $u$ and $v$ in $T$ (cf. Theorem \[t2.9\]).

\[p3.5\] The following statements are equivalent for every labeled tree $T = T(l)$.

1.  \[p3.5:s1\] The function $d_l$ is an ultrametric on $V(T)$.

2.  \[p3.5:s2\] The function $d_l$ is a metric on $V(T)$.

3.  \[p3.5:s3\] The inequality $$\label{p3.5:e1}
    \max\{l(u_1), l(v_1)\} > 0$$ holds for every $\{u_1, v_1\} \in E(T)$.

$\ref{p3.5:s1} \Rightarrow \ref{p3.5:s2}$. It follows directly from the definitions of metrics and ultrametrics.

$\ref{p3.5:s2} \Rightarrow \ref{p3.5:s3}$. Let \[p3.5:s2\] hold and let $\{u_1, v_1\} \in E(T)$. Then $u_1 \neq v_1$ holds because $T$ has no loops. Since $d_l$ is a metric, from 