Improved Bounds for the Expected Behaviour of AVL Trees

In this paper we improve previous bounds on expected measures of AVL trees by using fringe analysis. A new way of handling larger tree collections that are not closed is presented. An inherent diiculty posed by the transformations necessary to keep the AVL tree balanced makes its analysis diicult when using fringe analysis methods. We derive a technique to cope with this diiculty obtaining the exact solution for fringe parameters even when unknown probabilities are involved. We show that the probability of a rotation in an insertion is between 0.37 and 0.73 (and seems to be less than 0.56), that the fraction of balanced nodes is between 0.56 and 0.78, and that the expected number of comparisons in a search seems to be at most 12% more than in the complete balanced tree.