  We study two models of discrete height functions, that is, models of random
integer-valued functions on the vertices of a tree. First, we consider the
random homomorphism model, in which neighbours must have a height difference of
exactly one. The local law is uniform by definition. We prove that the height
variance of this model is bounded, uniformly over all boundary conditions (both
in terms of location and boundary heights). This implies a strong notion of
localisation, uniformly over all extremal Gibbs measures of the system. For the
second model, we consider directed trees, in which each vertex has exactly one
parent and at least two children. We consider the locally uniform law on height
functions which are monotone, that is, such that the height of the parent
vertex is always at least the height of the child vertex. We provide a complete
classification of all extremal gradient Gibbs measures, and describe exactly
the localisation-delocalisation transition for this model. Typical extremal
gradient Gibbs measures are localised also in this case. Localisation in both
models is consistent with the observation that the Gaussian free field is
localised on trees, which is an immediate consequence of transience of the
random walk.
