of isomorphism of trees.

\[d1.5\] Let $T_1$ and $T_2$ be trees. A bijection $f\colon V(T_1)\to V(T_2)$ is an *isomorphism* of $T_1$ and $T_2$ if $$\label{d1.5:e1}
(\{u,v\} \in E(T_1)) \Leftrightarrow (\{f(u),f(v)\} \in E(T_2))$$ is valid for all $u$, $v \in V(T_1)$. The trees $T_1$ and $T_2$ are *isomorphic* if there exists an isomorphism $f\colon V(T_1) \to V(T_2)$.

Let $T_1 = T_1(r_1)$ and $T_2 = T_2(r_2)$ be rooted trees. Then $T_1$ and $T_2$ are *isomorphic as rooted trees* if there is a bijection $f\colon V(T_1) \to V(T_2)$ such that $f(r_1) = r_2$ and is valid for all $u$, $v \in V(T_1)$.

For the case of labeled trees the above definition must be modified as follows.

\[d1.6\] Let $T_i=T_i(l_i)$ be labeled trees with the labelings $l_i\colon V(T_i)\to \mathbb{R}^{+}$, $i = 1$, $2$. An isomorphism $f\colon V(T_1) \to V(T_2)$ of the trees $T_1$ and $T_2$ is an isomorphism of the labeled trees $T_1(l_1)$ and $T_2(l_2)$ if the equality $$\label{d1.6e1}
l_2(f(v))=l_1(v)$$ holds for every $v \in V(T_1)$.

Analogously, for weighted trees $T_1 = T_1(w_1)$ and $T_2 = T_2(w_2)$ an isomorphism $f \colon V(T_1) \to V(T_2)$ of $T_1$ and $T_2$ is an isomorphism of $T_1(w_1)$ and $T_2(w_2)$ if the equality $$\label{d1.6e2}
w_2(\{f(u), f(v)\}) = w_1(\{u, v\})$$ 