Second phase changes in random m-ary search trees and generalized quicksort: convergence rates

We study the convergence rate to normal limit law for the space requirement of random m-ary search trees. While it is known that the random variable is asymptotically normally distributed for 3 ≤ m ≤ 26 and that the limit law does not exist for m > 26, we show that the convergence rate is O(n) for 3 ≤ m ≤ 19 and is O(n), where 4/3 < α < 3/2 is a parameter depending on m for 20 ≤ m ≤ 26. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the Berry-Esseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived. Abbreviated title. Phase changes in search trees.