Mathematical analysis of the average behavior of the AVL balanced tree insertion algorithm has not ever been done completely. As the first step toward this analysis, we have proposed an approximate analysis based on the assumption that all AVL balanced trees with a given number of nodes and height are constructed with equal probability. In this paper, we propose a new analysis of the AVL balanced tree insertion algorithm in conformity with that all n! permutations of n keys occur with equal probability. First we derive the formulae to enumerate the number of permutations constructing the AVL balanced trees with a given number of nodes and height. Then, we propose the formulae to evaluate the average behavior of the AVL balanced tree insertion algorithm and show some results from the proposed formulae.