  Search trees are fundamental data structures in computer science. We study
functionals on random search trees that satisfy recurrence relations of a
simple additive form. Many important functionals including the space
requirement, internal path length, and the so-called shape functional fall
under this framework. Our goal is to derive asymptotics of moments and identify
limiting distributions of these functionals under two commonly studied
probability models -- the random permutation model and the uniform model.
  For the random permutation model, our approach is based on establishing
transfer theorems that link the order of growth of the input into a particular
deterministic recurrence to the order of growth of the output. For the uniform
model, our approach is based on the complex-analytic tool of singularity
analysis. To facilitate a systematic analysis of these additive functionals we
extend singularity analysis, a class of methods by which one can translate on a
term-by-term basis an asymptotic expansion of a functional around its dominant
singularity into a corresponding expansion for the Taylor coefficients of the
function. The most important extension is the determination of how
singularities are composed under the operation of Hadamard product of analytic
power series.
  The transfer theorems derived are used in conjunction with the method of
moments to establish limit laws for m-ary search trees under the random
permutation model. For the uniform model on binary search trees, the extended
singularity analysis toolkit is employed to establish the asymptotic behavior
of the moments of a wide class of functionals. These asymptotics are used,
again in conjunction with the method of moments, to derive limit laws.
