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Random binary search trees, b-ary search trees, median-of-(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we offer a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a law of large numbers for the height.

Let H. be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of I, . . . , n. It is shown that HJog n + c = 4.3 I 107 . . . in probability as n + 00, where c is the unique solution of c log((2e)lc) = 1, c 2 2. Also, for all p > 0, lim,,E(H$)/ log% = cp. Finally, it is proved that &/log n --, c* = 0.3733 . . . , in probability, where c* is defined by c log((2e)lc) = 1, c 5 1, and .S, is the saturation level of the same tree, that is, the number of full levels in the tree.