Abstract
Random binary search trees are obtained by recursively inserting the elements of a uniformly random permutation σ of into a binary search tree data structure. Devroye (J. Assoc. Comput. Mach. 33 (1986) 489–498) proved that the height of such trees is asymptotically of order , where is the unique solution of with . In this paper, we study the structure of binary search trees built from Mallows permutations. A permutation is a random permutation of whose probability is proportional to , where . This model generalizes random binary search trees, since permutations with are uniformly distributed. The laws of and are related by a simple symmetry (switching the roles of the left and right children), so it suffices to restrict our attention to .
We show that, for , the height of is asymptotically in probability. This yields three regimes of behaviour for the height of , depending on whether tends to zero, tends to infinity or remains bounded away from zero and infinity. In particular, when tends to zero, the height of is asymptotically of order , like it is for random binary search trees. Finally, when tends to infinity, we prove stronger tail bounds and distributional limit theorems for the height of .
Funding Statement
During the preparation of this research, LAB was supported by an NSERC Discovery Grant and an FRQNT Team Grant, and BC was supported by an ISM Graduate Scholarship.
Acknowledgements
Both authors would like to thank an anonymous referee whose feedback both greatly improved the presentation of the paper and allowed us to shorten several of the proofs. BC also wishes to thank Ms. Legrand for supporting and encouraging his interest in mathematics.
Citation
Louigi Addario-Berry. Benoît Corsini. "The height of Mallows trees." Ann. Probab. 49 (5) 2220 - 2271, September 2021. https://doi.org/10.1214/20-AOP1503
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