Abstract
We study the number of records in random split trees on $n$ randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically $1$-stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees.
Highlights
We study the number of records in random split trees which were introduced by Devroye [2]
We recently showed that Janson’s approach could be applied for the random binary search tree [6]
In split trees on the other hand most vertices are close to the depth ∼ c ln n, where c is a constant; for the binary search tree that we investigated in [6] this depth is ∼ 2 ln n (e.g. [3])
Summary
We study the number of records in random split trees which were introduced by Devroye [2]. Given a rooted tree T with n nodes, let each vertex v have a random value λv attached to it, and assume that these values are i.i.d. with a continuous distribution. We say that a vertex v is a leaf in a split tree if the node itself holds at least one ball but no descendants of v hold any balls. When all the N balls have been distributed we end up with a split tree with a finite number of nodes which we denote by the parameter n. (i) If v is not a leaf, choose child i with probability Vi and recursively add the ball to the subtree rooted at child i, by the rules given in steps (i), (ii) and (iii).
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