  The parabolic Anderson model is the heat equation with some extra spatial
randomness. In this paper we consider the parabolic Anderson model with i.i.d.
Pareto potential on a critical Galton-Watson tree conditioned to survive. We
prove that the solution at time $t$ is concentrated at a single site with high
probability and at two sites almost surely as $t \to \infty$. Moreover, we
identify asymptotics for the localisation sites and the total mass, and show
that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a
certain functional of $v$. The main difference with earlier results on
$\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex
degrees within the tree, and make the role of the degrees precise.
