  In [1] a detailed analysis was given of the large-time asymptotics of the
total mass of the solution to the parabolic Anderson model on a supercritical
Galton-Watson random tree with an i.i.d. random potential whose marginal
distribution is double-exponential. Under the assumption that the degree
distribution has bounded support, two terms in the asymptotic expansion were
identified under the quenched law, i.e., conditional on the realisation of the
random tree and the random potential. The second term contains a variational
formula indicating that the solution concentrates on a subtree with minimal
degree according to a computable profile. The present paper extends the
analysis to degree distributions with unbounded support. We identify the
weakest condition on the tail of the degree distribution under which the
arguments in [1] can be pushed through. To do so we need to control the
occurrence of large degrees uniformly in large subtrees of the Galton-Watson
tree.
