This monograph is an attempt to provide some useful mathematical foundations for the design of interpreters for functional programming languages such as LISP. These languages are often described by sets of equations. One way to attack the problem of interpreting is to study ways of using equations to compute simple output expressions from input expressions. Such computations are studied here in terms of an abstract formalism called a Subtree Replacement System. With the hints given in Example 17 and Appendix A, a good programmer should be able to design efficient implementations of the abstract computations in a pointer manipulation system with garbage collection, and all of the mathematical results are intended to give useful information on the correctness, termination and efficiency properties of such implementations.