  We study the total mass of the solution to the parabolic Anderson model on a
regular tree with an i.i.d. random potential whose marginal distribution is
double-exponential. In earlier work we identified two terms in the asymptotic
expansion for large time of the total mass under the quenched law, i.e.,
conditional on the realisation of the random potential. In the present paper we
do the same for the annealed law, i.e., averaged over the random potential. It
turns out that the annealed expansion differs from the quenched expansion. The
derivation of the annealed expansion is based on a new approach to control the
local times of the random walk appearing in the Feynman-Kac formula for the
total mass. In particular, we condition on the backbone to infinity of the
random walk, truncate and periodise the infinite tree relative to the backbone
to obtain a random walk on a finite subtree with a specific boundary condition,
employ the large deviation principle for the empirical distribution of Markov
renewal processes on finite graphs, and afterwards let the truncation level
tend to infinity to obtain an asymptotically sharp asymptotic expansion.
