Heurlsttcs are considered which attempt to maintain a binary search tree in a near optimal form, assuming that elements are requested with fixed, but unknown, independent probabilities. A" move to root'" heuristic is shown to yield an expected search time within a constant factor of that of an optimal static binary search tree. On the other hand, a closely related" simple exchange" technique is shown not to have this property. The rate of convergence of the move to root heuristic is discussed Also considered is the more general case m whmh elements not in the tree may have nonzero probabihty of being requested.