$v \in V(T)$ are distinct, then we say that $v$ is a *successor* of $u$ whenever $u \in V(P)$, where $P$ is the path joining $v$ and $r$. A successor $v$ of $u$ is a *direct successor* of $u$ if $\{u, v\} \in E(T)$.

The following theorem is a simple modification of Theorem 2.7 [@DP2019PNUAA].

\[t1.7\] Let $T=T(r, l)$ be a labeled rooted tree. Then the following two conditions are equivalent:

1.  \[t1.7:c1\] For every $u \in V(T)$ we have $\delta^+(u)\neq 1$ and $$(\delta^+(u) =0) \Leftrightarrow (l(u)=0)$$ and, in addition, the inequality $l(v) < l(u)$ holds whenever $v$ is a direct successor of $u$.

2.  \[t1.7:c2\] There is a finite, nonempty ultrametric space $(X,d)$ such that $$T_X \simeq T(r, l).$$

Recall that a *phylogenetic tree* is an unrooted tree $T$, whose inner nodes have degree at least three, together with a labeling defined on the set of leaves of $T$ (see, for example, [@SS2009MH]). Using Theorem \[t13\] and above described procedure of construction of representing trees we can simply prove the following result.

\[p2.13\] The following statements are equivalent for every tree $T$.

1.  \[p2.13:s1\] There is a phylogenetic tree $T_1$ such that $T$ and $T_1$ are isomorphic as free 