a single point, then their union is a geodesic segment.

In the first definition, the second condition is automatic if $2\Lambda=\Lambda$, which is the case for $\Lambda={{\mathbb{R}}}^n$.

It follows from the definition that there is a unique geodesic joining a given pair of points in a $\Lambda$–tree.

Clearly, $\Lambda$ itself is $\Lambda$–tree. Another simple example of a $\Lambda$–tree is the vertex set $V(S)$ of a simplicial tree $S$: $V(S)$ endowed with the combinatorial distance is a ${{\mathbb{Z}}}$–tree.

Killing infinitesimals and extension of scalars
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The following two operations are usually known as the *base change functor*.

#### Killing infinitesimals

Consider $\Lambda_0\subset \Lambda$ a convex subgroup (a set of infinitesimals), and let ${\overline{\Lambda}}=\Lambda/\Lambda_0$. If $\Lambda={{\mathbb{R}}}^n$, we will usually take $\Lambda_0={{\mathbb{R}}}^{n-1}$, so that ${\overline{\Lambda}}\simeq{{\mathbb{R}}}$. Consider a $\Lambda$–metric space $E$. Then the relation $\sim$ defined by $x\sim y \Leftrightarrow d(x,y)\in \Lambda_0$ is an equivalence relation on $E$, and the $\Lambda$–metric on $E$ provides a natural ${\overline{\Lambda}}$–metric on $E/\sim$. We say that ${\overline{E}}=E/\sim$ is obtained from $E$ by *killing infinitesimals*. Clearly, if $T$ is a $\Lambda$–tree, then ${\overline{T}}$ is a ${\overline{\Lambda}}$–tree. Thus, killing infinitesimals in an ${{\mathbb{R}}}^n$–tree $T$ provides an ${{\mathbb{R}}}$–tree ${\overline{T}}$. By extension, we will often denote ${{\mathbb{R}}}$–trees with a bar.

#### Extension 