Constructing Red−Black Trees

This paper explores the structure of red-black trees by solv ing an apparently simple problem: given an ascending sequence of elements, construct, in linear tim e, a red-black tree that contains the elements in symmetric order. Several extreme red-black tree shapes are ch acterized: 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.