  We define and analyze random quantum walks on homogeneous trees of degree
$q\geq 3$. Such walks describe the discrete time evolution of a quantum
particle with internal degree of freedom in $\C^q$ hopping on the neighboring
sites of the tree in presence of static disorder. The one time step random
unitary evolution operator of the particle depends on a unitary matrix $C\in
U(q)$ which monitors the strength of the disorder. We prove for any $q$ that
there exist open sets of matrices in $U(q)$ for which the random evolution has
either pure point spectrum almost surely or purely absolutely continuous
spectrum, thereby showing the existence of a spectral transition driven by
$C\in U(q)$. For $q\in\{3,4\}$, we establish properties of the spectral diagram
which provide a description of the spectral transition.
