We study deterministic (non-deterministic) algorithms define by mean of confluent (resp. non-confluent) rewrite system admitting polynomial interpretation termination proofs. Data structures of the algorithms include strings, lists and trees. We classify them according to the interpretations of constructors This leads to the definition of six classes, which turn out to be exactly the deterministic (non-deterministic) poly-time, linear exponential-time and doubly linear exponential time computable functions when the class is based on confluent (resp. non-confluent) systems. Next, we demonstrate that functions with exponential interpretation termination proofs are super-elementary.