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A weakly 1-stable distribution for the number of random records and cuttings in split trees. (English) Zbl 1213.05037
Summary: 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 the 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 in C. Holmgren [“Random records and cuttings in binary search trees,” Comb. Probab. Comput. 19, No. 3, 391–424 (2010; Zbl 1215.05162), which is one specific case of split trees. Other important examples of split trees include $$m$$-ary search trees, quad trees, medians of $$(2k+1)$$-trees, simplex trees, tries, and digital search trees.

##### MSC:
 05C05 Trees 05C80 Random graphs (graph-theoretic aspects) 68W40 Analysis of algorithms 68P10 Searching and sorting 68R10 Graph theory (including graph drawing) in computer science 60C05 Combinatorial probability 60F05 Central limit and other weak theorems 68P05 Data structures
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