Reviewer: Don Goelman

Random binary search trees are the subclass of binary search trees (BSTs) built using only random insertions. That is, there is the same probability for the inserted key to fall into any of the j intervals defined by the j-1 keys already present in the tree. In this paper, the authors define and analyze structures they call randomized binary search trees (RBSTs). RBSTs are those trees produced by the specific insertion and deletion algorithms presented in the paper. Since the authors prove that RBSTs are always random BSTs, they can apply results known to hold for random BSTs. In particular, the expected cost of searches and updates is O log n , where n is the size of the tree. The insertion algorithm, part of which includes an implementation of Stephenson's algorithm, is shown to preserve random BSTs. In fact, while the standard insertion algorithm of a random permutation of keys into an empty BST yields a random BST, the algorithm in this paper yields a random BST on a fixed permutation as well. The deletion algorithm presented here also preserves the random property, as well as any arbitrary sequence of insertions and deletions, starting from an initially empty tree. Some interesting related discussions center on performance analysis and implementation (including consideration of node sizes, given auxiliary fields that might be needed), set operations (difference, union, and intersection), and implications for self-adjusting strategies (keys sought frequently should be near the root). The algorithms are illustrated with good examples and pictorial traces. Besides standard proofs, the authors use a formal algebraic diagrammatic approach following Martinez and Messeguer, here adapted to randomized algorithms. These results are compared with related randomized structures: treaps and skiplists. Both the research community and practitioners selecting appropriate data structures will find this paper useful and interesting.