We extend results about heights of random trees (Devroye, 1986(Devroye, , 1987(Devroye, , 1998b. In this paper, a general split tree model is considered in which the normalized subtree sizes of nodes converge in distribution. The height of these trees is shown to be in probability asymptotic to c log n for some constant c. We apply our results to obtain a law of large numbers for the height of all polynomial varieties of increasing trees (Bergeron et al., 1992).

We consider a model of random trees similar to the split trees of Devroye [30] in which a set of items is recursively partitioned. Our model allows for more flexibility in the choice of the partitioning procedure, and has weighted edges. We prove that for this model, the height H n of a random tree is asymptotic to c log n in probability for a constant c that is uniquely characterized in terms of multivariate large deviations rate functions. This extension permits us to obtain the height of pebbled tries, pebbled ternary search tries, d-ary pyramids, and to study geometric properties of partitions generated by k-d trees. The model also includes all polynomial families of increasing trees recently studied by Broutin, Devroye, McLeish, and de la Salle [17].

Let S = {s 1 , . . . , s n } be a set of points in the plane. The Oja depth of a query point θ with respect to S is the sum of the areas of all triangles (θ, s i , s j ). This depth may be computed in O(n log n) time in the RAM model of computation. We show that a matching lower bound holds in the algebraic decision tree model. This bound also applies to the computation of the Oja gradient, the Oja sign test, and to the problem of computing the sum of pairwise distances among points on a line.

We present an algorithm for hashing αn elements into a table with n separate chains that requires O(1) deterministic worst-case insert time, and O(1) expected worst-case search time for constant α. We exploit the connection between two-way chaining and random graph theory in our techniques.

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