a distance $d(v,w)$ on the set of its vertices, by adding the lengths of edges that appear in the unique path connecting $v$ and $w$ in the skeleton (which we still denote by $[[v,w]]$). The height of a vertex is its distance to the root. We let $\mathbb{T}_{k}$ be the set of trees with edge-lengths whose skeleton is in $\mathbf{T}_{k}$. For $\vartheta \in
\mathbb{T}_{k}$, let $e_{\mathrm{root}}$ be the length of the unique edge connected to the root, and for $e<e_{\mathrm{root}}$ write $\vartheta -e $ for the tree with edge-lengths that has same skeleton and same edge-lengths as $\vartheta $, but for the edge pointing outward the root which is assigned length $e_{\mathrm{root}}-e$.

We also define an operation $\mathtt{MERGE}$ as follows. Let $n\geq2$ and take $\vartheta_1,\vartheta_2,\ldots,\vartheta_n$ respectively in $\mathbb{T}_{k_1},\mathbb{T}_{k_2},\ldots,\mathbb{T}_{k_n}$, with leaves $(L^1_i,1\leq
i\leq k_1),(L^2_i,1\leq i\leq k_2),\ldots,(L^n_i,1\leq i\leq k_n)$ respectively. Let also $e>0$. The tree with edge-lengths $\mathtt{MERGE}((\vartheta_1,\ldots,\vartheta_n);e)\in \mathbb{T}_{\sum_i k_i}$ is defined by merging together the roots of $\vartheta_1,\ldots,\vartheta_n$ into a single vertex $\bullet$, and by drawing a new edge $\mathrm{root}\to\bullet$ with length $e$.

Last, for $\vartheta\in \mathbb{T}_k$ and $i$ vertices $v_1,\ldots,v_i$, define the subtree spanned by the root and $v_1,\ldots,v_i$ as follows. For every $p\neq q$, let $b(v_p,v_q)$ be the branchpoint 