In this paper we present randomized algorithms over binary search trees such that: a) the insertion of a set of keys, in any fixed order, into an initially empty tree always produces a random binary search tree; b) the deletion of any key from a random binary search tree results in a random binary search tree; c) the random choices made by the algorithms are based upon the sizes of the subtrees of the tree; this implies that we can support accesses by rank without additional storage requirements or modification of the data structures; and d) the cost of any elementary operation, measured as the number of visited nodes, is the same as the expected cost of its standard deterministic counterpart; hence, all search and update operations have guaranteed expected cost O(log n), but now irrespective of any assumption on the input distribution.