Limit Laws for Sums of Functions of Subtrees of Random Binary Search Trees

We consider sums of functions of subtrees of a random binary search tree, and obtain general laws of large numbers and central limit theorems. These sums correspond to random recurrences of the quicksort type, Xn L = XIn+X ′ n−1−In+Yn, n ≥ 1, where In is uniformly distributed on {0, 1, . . . , n− 1}, Yn is a given random variable, Xk L = X ′ k for all k, and given In, XIn and X ′ n−1−In are independent. Conditions are derived such that (Xn−μn)/σ √ n L → N (0, 1), the normal distribution, for some finite constants μ and σ.