  We study survival properties of inhomogeneous Galton-Watson processes. We
determine the so-called branching number (which is the reciprocal of the
critical value for percolation) for these random trees (conditioned on being
infinite), which turns out to be an a.s.\ constant. We also shed some light on
the way the survival probability varies between the generations. When we
perform independent percolation on the family tree of an inhomogeneous
Galton-Watson process, the result is essentially a family of inhomogeneous
Galton-Watson processes, parametrized by the retention probability $p$. We
provide growth rates, uniformly in $p$, of the percolation clusters, and also
show uniform convergence of the survival probability from the $n$-th level
along subsequences. These results also establish, as a corollary, the
supercritical continuity of the percolation function. Some of our results are
generalisations of results by Lyons (1992).
