Limit laws for functions of fringe trees for binary search trees and random recursive trees

Abstract

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings. As a consequence, we give simple new proofs of the fact that the number of fringe trees of size $ k=k_n $ in the binary search tree or in the random recursive tree (of total size $ n $) has an asymptotical Poisson distribution if $ k\rightarrow\infty $, and that the distribution is asymptotically normal for $ k=o(\sqrt{n}) $. Furthermore, we prove similar results for the number of subtrees of size $ k $ with some required property $ P $, e.g., the number of copies of a certain fixed subtree $ T $. Using the Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. <br /><br />We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of $ \ell $-protected nodes in a binary search tree or in a random recursive tree.