of isomorphism to simplify the statements of future results.

The concept of equivalent isomorphisms (= equivalent definitions of isomorphism) can be more satisfactory described on the basis of category theory, but this is not the subject of the paper.

Weights and labelings on unrooted trees {#sec3}
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Let $G(w)$ be a weighted graph. The weight $w \colon E(G) \to \mathbb{R}^{+}$ is *strictly positive* if $w(e) > 0$ holds for every $e \in E(G)$.

Let $T = T(w)$ be a weighted tree with the strictly positive weight $w$. The function $d_{w} \colon V(T) \times V(T) \to \mathbb{R}^{+}$ defined as $$\label{e6.7}
d_{w}(u, v) = \begin{cases}
0, & \text{if $u = v$}\\
\sum_{e \in E(P)} w(e), & \text{if $u \neq v$},
\end{cases}$$ where $P$ is the unique path joining $u$ and $v$ in $T$, is a metric on $V(T)$.

If $G = G(w)$ is a connected, weighted graph with the strictly positive weight $w \colon E(G) \to \mathbb{R}^{+}$, then the weighted *shortest-path metric* is the mapping $\rho_w \colon V(G) \times V(G) \to \mathbb{R}^{+}$ defined as $$\label{e3.2}
\rho_w(u, v) = \begin{cases}
0, & \text{if } u = v\\
\min_{P \in \mathcal{P}_{u, v}} \sum_{e \in P} w(e), & \text{if } u \neq v,
\end{cases}$$ where 