  In a recent paper by A. Backhausz, B. Gerencs\'er and V. Harangi, it was
shown that factors of independent identically distributed random processes
(IID) on trees obey certain geometry-driven inequalities. In particular, the
mutual information shared between two vertices decays exponentially, and there
is an explicit bound for this decay. In this note we show that all of these
inequalities could be verbatim translated to the setting of factors of
processes driven by unique Gibbs measures. As a consequence, we show that
correlations decay exponentially for unique Gibbs measures on trees.
