is *geodesic* if any two points are joined by a geodesic segment. We will denote by $[x,y]$ a geodesic segment between two points in $E$ (which, in this generality, might be non-unique).

Note that even in a set $\Lambda$ like ${{\mathbb{R}}}^n$, the upper bound is not always defined so one cannot easily define a $\Lambda$–valued diameter (see however [@Chi_book p.99] for a notion of diameter as a interval in $\Lambda$). Nevertheless, we will say that a subset $F$ of a ${{\mathbb{R}}}^n$–metric space $E$ is *infinitesimal* if the distance between any two points of $F$ is infinitesimal. Similarly, we define the *magnitude* of $F$ as the smallest $p\leq n$ such that the distance between any two points of $F$ has magnitude at most $p$.

We give two equivalent definitions of a $\Lambda$–tree. The equivalence is proved for instance in [@Chi_book Lemma 2.4.3, p.71].

A $\Lambda$–tree $T$ is a geodesic $\Lambda$–metric space such that

$T$ is $0$–hyperbolic in the following sense: $$\forall x,y,u,v\in T,\  d(x,y)+d(u,v)\leq \max\{d(x,u)+d(y,v),d(x,v)+d(y,u)\}$$

$\forall x,y,z\in T,\ d(x,y)+d(y,z)-d(x,z) \in 2\Lambda$

Equivalently, a geodesic $\Lambda$–metric space is a $\Lambda$–tree if

the intersection of any two geodesic segments sharing a common endpoint is a geodesic segment

if two geodesic segments intersect in 