We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If h∗ and u∗ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that h∗<2u−−√∗ in a broad enough set-up, but provide an example where 0=h∗=u∗ occurs. We also obtain some sufficient conditions ensuring that h∗>0.
