A functional limit theorem for the profile of search trees

We study the profile X-n,X-k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile X-n,X-k/EXn,k for k = [alpha logn] in a certain range of alpha. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.