We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex stabilizers and some more general, finite or infinite, edge stabilizers that we call highly core-free. We study the notion of highly core-free subgroups and give some examples. In the case of a free product amalgamated over a highly core-free subgroup and an HNN extension with a highly core-free base group we obtain a genericity result for faithful and highly transitive actions.