  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 $h_* < \sqrt{2u}_*$ in a broad enough set-up, but provide an example
where $0 = h_* = u_*$ occurs. We also obtain some sufficient conditions
ensuring that $h_* > 0$.
