This paper explores the structure of red-black trees by solving an apparently simple problem: given an ascending sequence of elements, construct a red-black tree which contains the elements in symmetric order. Several extreme red-black tree shapes are characterized: trees of minimum and maximum height, trees with a minimal and with a maximal proportion of red nodes. These characterizations are obtained by relating tree shapes to various number systems. In addition, connections to left-complete trees, AVL trees, and half-balanced trees are highlighted. 1 Introduction Red-black trees are an elegant search-tree scheme that guarantees O(log n) worstcase running time of basic dynamic-set operations. Recently, C. Okasaki (1998; 1999) presented an impressively simple functional implementation of red-black trees. In this paper we plunge deeper into the structure of red-black trees by solving an apparently simple problem: given an ascending sequence of elements, construct a red-black tree whic...

This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priority search queues can be used, for instance, to give a simple, purely functional implementation of Dijkstra's single-source shortest-paths algorithm.

Number systems serve admirably as templates for container types: a container object of size n is modelled after the representation of the number n and operations on container objects are modelled after their number-theoretic counterparts. Binomial queues are probably the first data structure that was designed with this analogy in mind. In this paper we show how to express these so-called numerical representations as higher-order nested datatypes. A nested datatype allows to capture the structural invariants of a numerical representation, so that the violation of an invariant can be detected at compile-time. We develop a programming method which allows to adapt algorithms to the new representation in a mostly straightforward manner. The framework is employed to implement three different container types: binary random-access lists, binomial queues, and 2-3 finger search trees. The latter data structure, which is treated in some depth, can be seen as the main innovation from a data-struct...