  In this paper we define a discrete generalized Laplacian with arbitrary real
power on a Cayley tree. This Laplacian is used to define a discrete generalized
Schr\"odinger operator on the tree. The case discrete fractional Schr\"odinger
operators with index $0 < \alpha < 2$ is considered in detail, and periodic
solutions of the corresponding fractional Schr\"odinger equations are
described. This periodicity depends on a subgroup of a group representation of
the Cayley tree. For any subgroup of finite index we give a criterion for
eigenvalues of the Schr\"odinger operator under which periodic solutions exist.
For a normal subgroup of infinite index we describe a wide class of periodic
solutions.
