On the richness of the collection of subtrees in random binary search trees

Abstract

The purpose of this paper is to settle two conjectures by Flajolet, Gourdon and Martinez (1996). We confirm that in a random binary tree on n nodes, the expected number of different subtrees grows indeed as Θ ( n log n ) . Secondly, if K is the largest integer such that all possible shapes of subtrees of cardinality less than or equal to K occur in a random binary search tree, then we show that K ~ log n log log n in probability.