it $\varnothing $. A *leaf* is a vertex which does not belong to $
[[\varnothing ,w[[:=\varphi _{\varnothing ,w}([0,d(\varnothing ,w)))$ for any vertex $w$. Call $\mathcal{L} (T)$ the set of leaves of $T$, and $
\mathcal{S}(T)=T\setminus \mathcal{L} (T)$ its skeleton. An $\mathbb{R}$ -tree is *leaf-dense* if $T$ is the closure of $\mathcal{L} (T)$. We also call *height* of a vertex $v$ the quantity $\mathrm{ht}
(v)=d(\varnothing ,v)$. Last, for $T $ an $\mathbb{R} $-tree and $a>0$, we let $a\otimes T$ be the $\mathbb{R}$-tree in which all distances are multiplied by $a$.

A continuum tree is a pair $(T,\mu )$ where $T$ is an $\mathbb{R}$-tree and $\mu $ is a probability measure on $T$, called the mass measure, which is non-atomic and satisfies $\mu (\mathcal{L}(T))=1$ and such that for every non-leaf vertex $w$, $\mu \{v\in T:[[\varnothing ,v]]\cap \lbrack \lbrack
\varnothing ,w]]=[[\varnothing ,w]]\}>0$. The set of vertices just defined is called the fringe subtree rooted at $w$. A CRT is a random variable $
\omega \mapsto (T(\omega ),\mu (\omega ))$ on a probability space $(\Omega ,
\mathcal{F},P)$ which values are continuum trees.

Notice that the definition of a continuum tree implies that the $\mathbb{R}$-tree $T$ satisfies certain extra properties, for example, its set of 