The profile of binary search trees. (English) Zbl 1012.60038

Summary: We characterize the limiting behavior of the number of nodes in level \(k\) of binary search trees \(T_n\) in the central region \(1.2\log n\leq k\leq 2.8\log n\). Especially we show that the width \(\overline V_n\) (the maximal number of internal nodes at the same level) satisfies \(\overline V_n \sim(n/ \sqrt{4\pi\log n})\) as \(n\to\infty\) a.s.


60F17 Functional limit theorems; invariance principles
05C05 Trees
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[1] Biggins, J.D. (1997). How fast does a general branching random walk spread? In Classical and Modern Branching Processes 19-39. Springer, New York. · Zbl 0873.60061
[2] Devroye, L. (1987). Branching processes in the analysis of the height of trees. Acta Inform. 24 277-298. · Zbl 0643.60065
[3] Drmota, M. and Hwang, H.K. (2001). Asymptotic expansions for second moments of the profile of binary search trees. Unpublished manuscript.
[4] Flajolet, P. and Odlyzko, A.M. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216-240. · Zbl 0712.05004
[5] Jabbour-Hattab, J. (2001). Martingales and large deviations for the binary search trees. Random Structure Algorithms 19, 112-127. · Zbl 0983.60034
[6] Joffe, A. Le Cam, L, and Neveu, J. (1973). Sur la loi des grands nombres pour des variables aléatoires attachées a un arbre dyadique. C. R. Acad. Sci. Paris. Ser. A 277 963-964. · Zbl 0292.60078
[7] Lynch, W. (1965). More combinatorial problems on certain trees.Computer J. 7 299-302. · Zbl 0136.38801
[8] Mahmoud, H.M. (1992). Evolution of Random Search Trees. Wiley, New York. · Zbl 0762.68033
[9] Moser, L. and Wyman, M. (1958). Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33 133-146. · Zbl 0081.28202
[10] Neveu, J. (1972). Martingales á temps discret. Masson and Cie, Paris. · Zbl 0235.60010
[11] Revuz, D. and Yor, M. (1990). Continuous Martingales and Brownian Motion. Springer, New York. · Zbl 1087.60040
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