  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 $m \leq 26$ and
typically periodic behavior if $m \geq 27$; (ii) for moderate toll sequences
[roughly, $t_n = \omega(n^{1 / 2})$ but $t_n = o(n)$] we have convergence to
non-normal distributions if $m \leq m_0$ (where $m_0 \geq 26$) and typically
periodic behavior if $m \geq m_0 + 1$; and (iii) for large toll sequences
[roughly, $t_n = \omega(n)$] we have convergence to non-normal distributions
for all values of m.
