that for all $x,y\in Y$, $[x,y]\subset Y$. A subtree is *non-degenerate* if it contains at least two points. One could think of endowing $\Lambda$, and $T$, with the order topology. However, this is usually not adapted. For instance: ${{\mathbb{R}}}^n$ is not connected with respect to this topology for $n>1$. This is why we need a special definition of a *closed* subtree. The definition coincides with the topological definition for ${{\mathbb{R}}}$–trees.

A subtree $Y\subset T$ is a *closed subtree* if the intersection of $Y$ with a segment of $T$ is either empty or a segment of $T$.

There is a natural projection on a closed subtree. Consider a base point $y_0\in Y$. Then for any point $x\in T$, there is a unique point $p\in Y$ such that $[y_0,x]\cap Y=[y_0,p]$. One easily checks that $p$ does not depend on the choice of the base point $y_0$ ($[p,x]$ is the *bridge* between $x$ and $Y$, see [@Chi_book]). The point $p$ is called the *projection* of $x$ on $Y$.

The existence of a projection is actually equivalent to the fact that the subtree $Y$ is closed. Be aware that a non-trivial proper convex subgroup of $\Lambda$ is never closed in $\Lambda$. In particular, the 