  We study Hecke algebras of groups acting on trees with respect to
geometrically defined subgroups. In particular, we consider Hecke algebras of
groups of automorphisms of locally finite trees with respect to vertex and edge
stabilizers and the stabilizer of an end relative to a vertex stabilizer,
assuming that the actions are sufficiently transitive. We focus on identifying
the structure of the resulting Hecke algebras, give explicit multiplication
tables of the canonical generators and determine whether the Hecke algebra has
a universal C*-completion. The paper unifies past algebraic and analytic
approaches by focusing on the common geometric thread.The results have
implications for the general theory of totally disconnected locally compact
groups.
