The simple random walk on a supercritical Galton–Watson
tree is transient when the tree is infinite. Moreover, there exist regeneration
times, that is, times when the walk crosses an edge for the first and the last
time. We prove that the distance and the range of the walk satisfy functional
central limit theorems under the annealed law GP. This result is a
consequence of estimates of the law of the first regeneration time $\tau_R$. We
show that there exist positive $c , c', \alpha$ and $\beta$ such that, for all
$n \geq 1$, $$\exp (-cn^{\alpha}) \leq GP[\tau_R \geq \exp (-c'
n^{\beta}).$