  Motivated by the Bruhat and Cartan decompositions of general linear groups
over local fields, 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 are enumerated. This enumeration is used to show that
every continuous homomorphism from the automorphism group of a label-regular
tree has closed range.
