An almost sure result for path lengths in binary search trees

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the 'quicksort distribution'. For the class Am of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in Am. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.