Automorphism groups of locally finite trees provide a large class of examples
of simple totally disconnected locally compact groups. It is desirable to
understand the connections between the global and local structure of such a
group. Topologically, the local structure is given by the commensurability
class of a vertex stabiliser; on the other hand, the action on the tree
suggests that the local structure should correspond to the local action of a
stabiliser of a vertex on its neighbours.