  We consider invasion percolation on Galton-Watson trees. On almost every
Galton-Watson tree, the invasion cluster almost surely contains only one
infinite path. This means that for almost every Galton-Watson tree, invasion
percolation induces a probability measure on infinite paths from the root. We
show that under certain conditions of the progeny distribution, this measure is
absolutely continuous with respect to the limit uniform measure. This confirms
that invasion percolation, an efficient self-tuning algorithm, may be used to
sample approximately from the limit uniform distribution. Additionally, we
analyze the forward maximal weights along the backbone of the invasion cluster
and prove a limit law for the process.
