We propose a definition of branching-type stationary stochastic processes on rooted trees. 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 spectral measures of branching-type stationary stochastic processes 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. This talk is based on a joint work with Zipeng WANG.