Constructing optimal search trees in optimal time

(a, b)-trees are an important class of search trees. They include 2-3 trees, 2-3-4 trees, and B-trees as subclasses. We show that a space-minimum (a, b)-tree is also height-minimum and present an optimal algorithm for constructing (a, b)-trees that are height-minimum and space-minimum. Given n keys, our algorithm constructs an (a, b)-tree with minimum height and lowest possible nodes. Our algorithm takes Θ(n) time if the keys in S are sorted and Θ(n log n) time if the keys are not sorted. We also discuss possible applications of our algorithm