Transfer Theorems And Asymptotic Distributional Results For M-Ary Search Trees

We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals 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. The 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. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (t(n)) [roughly, t(n) = O(n(1/2))] we have asymptotic normality if m <= 26 and typically periodic behavior if m >= 27; (ii) for moderate toll sequences [roughly, t(n) = omega(n(1/2)) but t(n) = o(n)] we have convergence to nonnormal distributions if m <= m(0) (where m(0) >= 26) and typically periodic behavior if m >= m(0) + 1; and (iii) for large toll sequences [roughly, t(n) = omega(n)] we have convergence to normormal distributions for all values of m. (c) 2004 Wiley Periodicals, Inc.