Let X be a locally finite tree and Aut(X) the locally compact group of all isometries of X. It is proved that a closed subgroup G of Aut(X) is amenable if and only if G has one of the following properties: (i) G fixes a vertex; (ii) G leaves invariant an edge; (iii) G fixes an end of X; (iv) G leaves invariant a pair of ends of X.
A locally compact group G is said to be a Kunze-Stein group if \(L^ p(G)*L^ 2(G)\subset L^ 2(G)\) for every \(1<p<2\). Let X be a homogeneous tree and G a subgroup of Aut(X) acting transitively on the vertices and on an open subset of the boundary of X. It is shown that G is either amenable or a Kunze-Stein group. The proofs depend on results on J. Tits [Essays on topology and related topics, 188-211 (1970; Zbl 0214.513)].