  We study closed subgroups $G$ of the automorphism group of a locally finite
tree $T$ acting doubly transitively on the boundary. We show that if the
stabiliser of some end is metabelian, then there is a local field $k$ such that
$\mathrm{PSL}_2(k) \leq G \leq \mathrm{PGL}_2(k)$. We also show that the
contraction group of some hyperbolic element is closed and torsion-free if and
only if $G$ is (virtually) a rank one simple $p$-adic analytic group for some
prime $p$. A key point is that if some contraction group is closed, then $G$ is
boundary-Moufang, meaning that the boundary $\partial T$ is a Moufang set. We
collect basic results on Moufang sets arising at infinity of locally finite
trees, and provide a complete classification in case the root groups are
torsion-free.
