${{\mathbb{R}}}$ (see for instance [@Chi_book Theorem 1.1.2])

If $\Lambda_0\subset \Lambda$ is a convex subgroup, then any element $\lambda_0\in\Lambda_0$ may be thought as infinitely small compared to an element $\lambda\in\Lambda\setminus\Lambda_0$ since for all $n\in{{\mathbb{N}}}$, $n\lambda_0\leq \lambda$. Therefore, we will say that an element in ${{\mathbb{R}}}^n$ is *infinitesimal* if it lies in the maximal proper convex subgroup of ${{\mathbb{R}}}^n$, which we casually denote by ${{\mathbb{R}}}^{n-1}$. Similarly, for $p\leq n$, we will identify ${{\mathbb{R}}}^p$ with the corresponding convex subgroup of ${{\mathbb{R}}}^n$. The *magnitude* of an element $\lambda\in{{\mathbb{R}}}^n$ is the smallest $p$ such that $\lambda\in {{\mathbb{R}}}^p$. Thus $\lambda\in{{\mathbb{R}}}^n$ is infinitesimal if and only if its magnitude is at most $n-1$.

Given a totally ordered abelian group $\Lambda$, $\Lambda{\otimes}{{\mathbb{Q}}}$ has a natural structure of a totally ordered abelian group by letting $\frac{\lambda}{n}\leq\frac{\lambda'}{n'}$ if and only if $n'\lambda\leq n\lambda'$.

#### $\Lambda$–metric spaces and $\Lambda$–trees

A *$\Lambda$–metric space* $(E,d)$ is a set $E$ endowed with a map $d\co E\times E{\rightarrow}\Lambda_{\geq 0}$ satisfying the three usual axioms of a metric: separation, symmetry and triangle inequality. The set $\Lambda$ itself is a $\Lambda$–metric space for the metric $d(a,b)=|a-b|=\max(a-b,b-a)\in\Lambda$. A *geodesic segment* in $E$ is an isometric map from a segment $[a,b]\subset\Lambda$ to a subset of $E$. A $\Lambda$–metric space 