  We consider stochastic processes with (or without) memory whose evolution is
encoded by a finite or infinite rooted tree. The main goal is to compare the
entropy rates of a given base process and a second one, to be considered as a
perturbation of the former. The processes are described by probability measures
on the boundary of the given tree, and by corresponding forward transition
probabilities at the inner nodes. The comparison is in terms of
Kullback-Leibler divergence. We elaborate and extend ideas and results of
B\"ocherer and Amjad. Our extensions involve length functions on the edges of
the tree as well as nodes with countably many successors. In particular, in the
last part, we consider trees with infinite geodesic rays and random
perturbations of a given process.
