Cauchy-Euler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for Cauchy-Euler equations and propose an asymptotic theory that covers almost all applications where Cauchy-Euler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.

Fringe analysis is a technique used to study the average behavior of search trees. In this paper we survey the main results regarding this technique, and we improve a previous asymptotic theorem. At the same time we present new developments and applications of the theory which allow improvements in several bounds on the behavior of search trees. Our examples cover binary search trees, AVL trees, 2-3 trees, and B-trees. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity ]: Nonnumerical Algorithms and Problems -- computations on discrete structures; sorting and searching; E.1 [Data Structures]; trees. Contents 1 Introduction 2 2 The Theory of Fringe Analysis 4 3 Weakly Closed Collections 9 4 Including the Level Information 11 5 Fringe Analysis, Markov Chains, and Urn Processes 13 This work was partially funded by Research Grant FONDECYT 93-0765. e-mail: rbaeza@dcc.uchile.cl 1 Introduction Search trees are one of the most used data structures t...

We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful search is proportional to the number of ascendants of the sought element. On the other hand, the probabilistic behavior of the number of descendants is relevant for the analysis of paged data structures and for the analysis of the performance of quicksort, when recursive calls are not made on small subfiles. We also consider the number of ascendants and descendants of a random node in a random search tree, i.e., the grand averages of the quantities mentioned above. We address these questions for standard binary search trees and for locally balanced search trees. These search trees were introduced by Poblete and Munro and are binary search trees such that each subtree of size 3 is balanced; in oth...