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...

In this paper we explore the relationship between asymmetries in deletion algorithms used in updating binary search trees, and the resulting long term behavior of the search trees. We show that even what would appear to be negligible asymmetric effects accumulate to cause long term degeneration. This persists even in the face of other effects that would appear to counteract the long term effects. On the other hand, eliminating the asymmetry completely seems to give us trees that have a smaller IPL than is expected for trees built by a random sequence of insertions. But even then there are surprises in that the backbone becomes longer than expected. 1 Introduction Binary search trees are one of the oldest and most frequently used data structures for solving the dictionary and other problems [2, 11, 6, 9]. The average case efficiency of these structures has been well studied, when only insertions are involved. The usual insertion algorithm simply inserts new values at the leaf...

We present a fourth-order fringe analysis for the expected behavior of 2-3 trees, which includes 97% of the elements in the tree. It is accomplished by exploiting the structure of the transition matrix. Our results improve a number of bounds, in particular the bounds on the expected number of nodes and the expected space utilization. We also study 2-3 trees built by using overflow techniques. 1 Introduction Fringe analysis was formally introduced by Yao in 1974 [Yao74, Yao78] as a method to analyze search trees that considers only the bottom part or fringe of the tree. From the behavior of the subtrees in the fringe, it is possible to obtain bounds on most complexity measures for the complete tree, as well as some exact results. Classical fringe analysis considers only insertions. The model assumes that the n! possible permutations of the n keys used as input are equally likely. A search tree built under this model is called a random tree. This is equivalent to saying that the n-th in...