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Automorphism groups of locally finite trees provide a large class of examplesof simple totally disconnected locally compact groups. It is desirable tounderstand the connections between the global and local structure of such agroup. Topologically, the local structure is given by the commensurabilityclass of a vertex stabiliser; on the other hand, the action on the treesuggests that the local structure should correspond to the local action of astabiliser of a vertex on its neighbours.  We study the interplay between these different aspects for the special classof groups satisfying Tits' independence property. We show that such a group hasfew open subgroups if and only if it acts locally primitively. Moreover, weshow that it always admits many germs of automorphisms which do not extend toautomorphisms, from which we deduce a negative answer to a question by GeorgeWillis. Finally, under suitable assumptions, we compute the full group of germsof automorphisms; in some specific cases, these turn out to be simple andcompactly generated, thereby providing a new infinite family of examples whichgeneralise Neretin's group of spheromorphisms. Our methods describe moregenerally the abstract commensurator group for a large family ofself-replicating profinite branch groups.

Simple locally compact groups acting on trees and their germs of automorphisms