Decomposition theorems for automorphism groups of trees

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.