Additive functionals on multiway search trees

We derive asymptotics of moments and limiting distributions, under the random permutation model on m-ary search trees on n keys, of functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the socalled shape functional fall under this framework. The limiting behavior of these functionals exhibit intriguing phase changes. For suitably “small” input (or toll) sequences we have asymptotic normality if m ≤ 26 and typically periodic behavior otherwise. For “moderate” toll sequences we have convergence to non-normal distributions if m ≤ m0 (where m0 ≥ 26) and typically periodic behavior otherwise. For “large” toll sequences we have convergence to non-normal distributions for all values of m. Recent research greatly sharpens the understanding of the periodic cases. For example, Chauvin and Pouyanne have shown that for m ≥ 27 fixed, the space requirement equals