E_2)$ for all $u$, $v \in V_1$.

3.  \[d2.11:s3\] $l_1(\{u, v\}) = l_2(\{f(u), f(v)\})$ for every $\{u, v\} \in E_1$

(See Definition 1 in [@HHH2006]).

Two definitions of graph isomorphism are equivalent if any two graphs are isomorphic w.r.t the first definition if and only if these graphs are isomorphic w.r.t. the second one. For the case when $$G_1 = (V_1, E_1, L_{V_1}, L_{E_1}, l_1) \quad \text{and} \quad G_2 = (V_2, E_2, L_{V_2}, L_{E_2}, l_2)$$ are trees, Definition \[d2.11\] is equivalent to:

-   Definition \[d1.5\] if $L_{V_1} \cup L_{E_1} = L_{V_2} \cup L_{E_2} = \{0\}$.

-   The definition of isomorphic rooted trees if $L_{V_1} \cup L_{E_1} = L_{V_2} \cup L_{E_2} = \{0, 1\}$ and there are the unique $v^{*}_1 \in V_1$ and $v^{*}_2 \in V_2$ such that $l_i(v^{*}_i) = 1$ and $l_i(e) = l_i(v) = 0$ whenever $e \in E_i$ and $v \in V_i \setminus \{v^{*}_i\}$, $i = 1$, $2$.

-   Definition \[d1.6\] if $L_{V_1} = L_{V_2} = \mathbb{R}^{+}$ and $L_{E_1} = L_{E_2} = \{0\}$.

Similarly we can find specifications of Definition \[d2.11\] which are equivalent to definitions of isomorphisms of phylogenetic trees, weighted trees, weighted rooted trees, and weighted labeled trees, but we prefer to use several independent definitions 