A Weakly 1-Stable Limiting Distribution for the Number of Random Records and Cuttings in Split Trees

We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 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, medians of $(2k+1)$-trees, simplex trees, tries and digital search trees.