  We study Gibbsian models of unbounded integer-valued spins on trees which
possess a symmetry under height-shift. We develop a theory relating boundary
laws to gradient Gibbs measures, which applies also in cases where the
corresponding Gibbs measures do not exist. Our results extend the classical
theory of Zachary beyond the case of normalizable boundary laws, which implies
existence of Gibbs measures, to periodic boundary laws. We provide a
construction for classes of tree-automorphism invariant gradient Gibbs measures
in terms of mixtures of pinned measures, whose marginals to infinite paths on
the tree are random walks in a q-periodic environment. Here the mixture measure
is the invariant measure of a finite state Markov chain which arises as a mod-q
fuzzy transform, and which governs the correlation decay. The construction
applies for example to SOS-models and discrete Gaussian models and delivers a
large number of gradient Gibbs measures. We also discuss relations of certain
gradient Gibbs measures to Potts and Ising models.
