  We show that many countable groups acting on trees, including free products
of infinite countable groups and surface groups, are isomorphic to dense
subgroups of isometry groups of bounded Urysohn spaces. This extends previous
results of the first and last author with Y. Stalder on dense subgroups of the
automorphism group of the random graph. In the unbounded case, we also show
that every free product of infinite countable groups arises as a dense subgroup
of the isometry group of the rational Urysohn space.
