  In earlier work by den Hollander, K\"onig, and dos Santos, the asymptotics of
the total mass of the solution to the parabolic Anderson model was studied on
an almost surely infinite Galton-Watson tree with an i.i.d. potential having a
double-exponential distribution. The second-order contribution to this
asymptotics was identified in terms of a variational formula that gives
information about the local structure of the region where the solution is
concentrated. The present paper extends this work to the degree-normalised
Laplacian. The normalisation causes the Laplacian to be non-symmetric and which
leads to different spectral properties. We find that the leading order
asymptotics of the total mass remains the same, while the second-order
correction coming from the variational formula is different. We also find that
the optimiser of the variational formula is again an infinite tree with minimal
degrees. Both of these results are shown to hold under much milder conditions
than for the regular Laplacian.
