  We introduce trap models on a finite volume $k$-level tree as a class of
Markov jump processes with state space the leaves of that tree. They serve to
describe the GREM-like trap model of Sasaki and Nemoto. Under suitable
conditions on the parameters of the trap model, we establish its infinite
volume limit, given by what we call a $K$-process in an infinite $k$-level
tree. From this we deduce that the $K$-process also is the scaling limit of the
GREM-like trap model on extreme time scales under a fine tuning assumption on
the volumes.
