  We analyse the spectral phase diagram of Schr\"odinger operators $ T +\lambda
V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a
random potential given by iid random variables. The main result is a criterion
for the emergence of absolutely continuous (ac) spectrum due to
fluctuation-enabled resonances between distant sites. Using it we prove that
for unbounded random potentials ac spectrum appears at arbitrarily weak
disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the
spectrum of $T$. Incorporating considerations of the Green function's large
deviations we obtain an extension of the criterion which indicates that, under
a yet unproven regularity condition of the large deviations' 'free energy
function', the regime of pure ac spectrum is complementary to that of
previously proven localization. For bounded potentials we disprove the
existence at weak disorder of a mobility edge beyond which the spectrum is
localized.
