  We propose a definition of branching-type stationary stochastic processes on
rooted trees and related definitions of hyper-positivity for functions on the
unit circle and functions on the set of non-negative integers. We then obtain
(1) a necessary and sufficient condition on a rooted tree for the existence of
non-trivial branching-type stationary stochastic processes on it, (2) a
complete criterion of the hyper-positive functions in the setting of rooted
homogeneous trees in terms of a variant of the classical Herglotz-Bochner
Theorem, (3) a prediction theory result for branching-type stationary
stochastic processes.
  As an unexpected application, we obtain natural hypercontractive inequalities
for Hankel operators with hyper-positive symbols.
