  We consider a random Schr\"odinger operator on the binary tree with a random
potential which is the sum of a random radially symmetric potential, $Q_r$, and
a random transversally periodic potential, $\kappa Q_t$, with coupling constant
$\kappa$. Using a new one-dimensional dynamical systems approach combined with
Jensen's inequality in hyperbolic space (our key estimate) we obtain a
fractional moment estimate proving localization for small and large $\kappa$.
Together with a previous result we therefore obtain a model with two Anderson
transitions, from localization to delocalization and back to localization, when
increasing $\kappa$. As a by-product we also have a partially new proof of
one-dimensional Anderson localization at any disorder.
