  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 $3\le m\le 13$
but becomes of higher order when $m\ge14$. Surprisingly, the corresponding
asymptotic correlation coefficient tends to zero when $3\le m\le 26$ 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 fringe-balanced binary search trees and quadtrees. The methods of proof
combine asymptotic transfer for the underlying recurrence relations with the
contraction method.
