Branching Processes and Their Applications in the Analysis of Tree Structures and Tree Algorithms

Abstract

We give a partial overview of some results from the rich theory of branching processes and illustrate their use in the probabilistic analysis of algorithms and data structures. The branching processes we discuss include the Galton-Watson process, the branching random walk, the Crump-Mode-Jagers process, and conditional branching processes. The applications include the analysis of the height of random binary search trees, random m-ary search trees, quadtrees, union-find trees, uniform random recursive trees and plane-oriented recursive trees. All these trees have heights that grow logarithmically in the size of the tree. A different behavior is observed for the combinatorial models of trees, where one considers the uniform distribution over all trees in a certain family of trees. In many cases, such trees are distributed like trees in a Galton-Watson process conditioned on the tree size. This fact allows us to review Cayley trees (random labeled free trees), random binary trees, random unary-binary trees, random oriented plane trees, and indeed many other species of uniform trees. We also review a combinatorial optimization problem first suggested by Karp and Pearl. The analysis there is particularly beautiful and shows the flexibility of even the simplest branching processes.