of $v_p$ and $v_q$, that is, the highest point in the tree that belongs to $[[\mathrm{root},v_p]]\cap [[\mathrm{root},v_q]]$. The spanned tree is the tree with edge-lengths whose vertices are the root, the vertices $v_1,\ldots,v_i$ and the branchpoints $b(v_p,v_q)$, $1\leq p\neq q\leq i$, and whose edge-lengths are given by the respective distances between this subset of vertices of the original tree.

Building the CRT
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Now for $B\subset \mathbb{N}$ finite, define $\mathcal{R}(B)$, a random variable with values in $\mathbb{T}_{\#B}$, whose leaf-labels are of the form $L_i$ for $i\in \mathbb{N}$ , as follows. Let $D_i=\inf\{t\geq
0:\{i\}\in\Pi(t)\}$ be the first time when $\{i\}$ “disappears”, i.e. is isolated in a singleton of $\Pi(t)$. For $B$ a finite subset of $\mathbb{N}$ with at least two elements, let $D_B=\inf\{t\geq 0:\#(B\cap \Pi(t))\neq 1\}$ be the first time when the restriction of $\Pi(t)$ to $B$ is non-trivial, i.e. has more than one block. By convention, $D_{\{i\}}=D_i$. For every $i\geq 1$, define $\mathcal{R}(\{i\})$ as a single edge $\mathrm{root}\to L_i$, and assign this edge the length $D_i$. For $B$ with $\#B\geq 2$, let $B_1,\ldots,B_i$ be the non-empty blocks of $B\cap\Pi(D_B)$, arranged in increasing order of least element, and define a tree $\mathcal{R}(B)$ recursively by $$\mathcal{R}(B)=\mathtt{MERGE}((\mathcal{R}(B_1)-D_B,\ldots,\mathcal{R}(B_i)-D_B);D_B).$$ Last, define $\mathcal{R}(k)=\mathcal{R}([k])$. Notice that by 