Split Trees – A Unifying Model for Many Important Random Trees of Logarithmic Height: A Brief Survey

Abstract

Split trees were introduced by Devroye [28] as a novel approach for unifying many important random trees of logarithmic height. They are interesting not least because of their usefulness as models of sorting algorithms in computer science; for instance the well-known Quicksort algorithm (introduced by Hoare [35, 36]) can be depicted as a binary search tree (which is one example of a split tree). A split tree of cardinality n is constructed by distributing n balls (which often represent data items) to a subset of nodes of an infinite tree. In [39], renewal theory was introduced by the author as a novel approach for studying split trees. This approach has proved to be highly useful for investigating such trees and has lead (often in combination with other methods) to several general results valid for all split trees. In this brief survey, we will present split trees, give an introduction to renewal theory in relation to split trees and describe some of the characteristics of split trees including results on the depths for the balls and nodes in the tree, the height (maximal depth) of the tree and the size of the tree in terms of the number of nodes; see [11, 17, 18, 28, 39]. Furthermore, we will briefly describe some of our later results for this large class of random trees, e.g. on the total path length [19], number of cuttings [12, 21, 38] and number of inversions (and more general permutations) [2, 20] as well as on the size of the giant and other components after bond percolation [11, 13].