This report is a contributed chapter to the Handbook of Theoretical Computer Science (North-Holland, 1990). Its aim is to describe the main mathematical methods and applications in the average-case analysis of algorithms and data structures. It comprises two parts: First, we present basic combinatorial enumerations based on symbolic methods and asymptotic methods with emphasis on complex analysis techniques (such as singularity analysis, saddle point, Mellin transforms). Next, we show how to apply these general methods to the analysis of sorting, searching, tree data structures, hashing, and dynamic algorithms. The emphasis is on algorithms for which exact "analytic models" can be derived.

. Lambda-Upsilon-Omega, \Upsilon\Omega , is a system designed to perform automatic analysis of well-defined classes of algorithms operating over "decomposable" data structures. It consists of an `Algebraic Analyzer' System that compiles algorithms specifications into generating functions of average costs, and an `Analytic Analyzer' System that extracts asymptotic informations on coefficients of generating functions. The algebraic part relies on recent methodologies in combinatorial analysis based on systematic correspondences between structural type definitions and counting generating functions. The analytic part makes use of partly classical and partly new correspondences between singularities of analytic functions and the growth of their Taylor coefficients. The current version \Upsilon\Omega 0 of \Upsilon\Omega implements as basic data types, term trees as encountered in symbolic algebra systems. The analytic analyzer can treat large classes of functions with explicit expressio...