A complexity calculus for classes of recursive search programs over tree structures

We study a restricted programming language over tree structures. For this language, we give systematic translation rules which map programs into complexity descriptors. The descriptors are in the form of generating functions of average costs. Such a direct approach avoids the recourse to recurrences; it therefore simplifies the task of analyzing algorithms in the class considered and permits analysis of structurally complex programs. It also allows for a clear discussion of analytic properties of complexity descriptors whose singularities are related to the asymptotic behavior of average costs. Algorithms that are analyzed in this way include: formal differentiation, tree matching, tree embedding and simplification of expressions in a diversity of contexts. Some general results relating (average case) complexity properties to structural properties of programs in the class can also be derived in this framework.