  Several types of term rewriting systems can be distinguished by the way their
rules overlap. In particular, we define the classes of prefix, suffix,
bottom-up and top-down systems, which generalize similar classes on words. Our
aim is to study the derivation relation of such systems (i.e. the reflexive and
transitive closure of their rewriting relation) and, if possible, to provide a
finite mechanism characterizing it. Using a notion of rational relations based
on finite graph grammars, we show that the derivation of any bottom-up,
top-down or suffix systems is rational, while it can be non recursive for
prefix systems.
