We study the effect of polynomial interpretation termination proofs of deterministic (resp. non-deterministic) algorithms defined by confluent (resp. non-confluent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of the constructors. This leads to the definition of six function classes which turn out to be exactly the deterministic (resp. non-deterministic) polynomial time, linear exponential time and linear doubly exponential time computable functions when the class is based on confluent (resp. non-confluent) rewrite systems. We also obtain a characterisation of the linear space computable functions. Finally, we demonstrate that functions with exponential interpretation termination proofs are super-elementary.