  We study the profile $X_{n,k}$ 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 $X_{n,k}/\mathbb{E}X_{n,k}$ for $k=\lfloor\alpha\log
n\rfloor$ 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.
