Dependence and phase changes in random m-ary search trees

We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all root-key distances or over all root-node distances) in random m-ary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when 3 6 m 6 13 but becomes of higher order when m > 14. Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when 3 6 m 6 26 but is periodically oscillating for larger m. Such a less anticipated phenomenon is not exceptional and we extend the results in two directions: one for more general shape parameters, and the other for other classes of random log-trees such as fringebalanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method. AMS 2010 subject classifications. Primary 60F05, 68Q25; secondary 68P05, 60C05, 05A16.