The subject of this work are random Schroedinger operators on regular rooted
tree graphs $\T$ with stochastically homogeneous disorder. The operators are of
the form $H_\lambda(\omega) = T + U + \lambda V(\omega)$ acting in
$\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic
potential, and $V(\omega)$ a random potential. This includes the only class of
homogeneously random operators for which it was proven that the spectrum of
$H_\lambda(\omega)$ exhibits an absolutely continuous (ac) component; a results
established by A. Klein for weak disorder, in case U=0 and $V(\omega)$ given by
iid random variables on $\T$. Our main contribution is a new method for
establishing the persistence of ac spectrum under weak disorder. The method
yields the continuity in the disorder parameter of the ac spectral density of
$H_\lambda(\omega)$ at $\lambda = 0$. The latter is shown to converge in the
$L^1$ sense over closed intervals in which $H_0$ has no singular spectrum. The
analysis extends to random potentials whose values at different sites need not
be independent, assuming only that their joint distribution is weakly
correlated across different tree branches.