Uniform distribution modulo one and binary search trees
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 415-424.

Any sequence $x={\left({x}_{k}\right)}_{k=1}^{\infty }$ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let ${H}_{n}\left(x\right)$ be the height of the tree generated by ${x}_{1},\cdots ,{x}_{n}.$ Obviously

 $\frac{logn}{log2}-1\le {H}_{n}\left(x\right)\le n-1.$
If the sequences $x$ are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists $c>0$ such that ${H}_{n}\left(x\right)\sim clogn$ as $n\to \infty$ for almost all sequences $x$. Recently Devroye and Goudjil have shown that if the sequences $x$ are Weyl sequences, i.e., defined by ${x}_{k}=\left\{\alpha k\right\},k=1,2,\cdots ,$ and $\alpha$ is distributed uniformly at random on $\left[0,1\right]$ then
 ${H}_{n}\left(x\right)\sim \left(12/{\pi }^{2}\right)lognloglogn$
as $n\to \infty$ in probability. In this paper we consider the class of all uniformly distributed sequences $x$, and we show that the only behaviour that is excluded by the equidistribution of $x$ is that of the worst case, i.e., for these $x$ we necessarily have ${H}_{n}\left(x\right)=o\left(n\right)$ as $n\to \infty$.

On peut construire à partir d’une suite $x={\left({x}_{k}\right)}_{k=1}^{\infty }$ de nombres distincts de l’intervalle [0,1] un arbre binaire en plaçant successivement ces nombres sur les noeuds selon un algorithme “gauche-droite” (cela revient à classer les nombres selon l’algorithme Quicksort). On note ${H}_{n}\left(x\right)$ la hauteur de l’arbre obtenu à partir des nombres ${x}_{1},\cdots ,{x}_{n}.$ Il est évident que

 $\frac{logn}{log2}-1\le {H}_{n}\left(x\right)\le n-1.$
Si la suite $x$ est obtenue comme valeurs de variables aléatoires indépendantes uniformes sur [0,1], alors on sait qu’il existe $c>0$ tel que ${H}_{n}\left(x\right)\sim clogn,\left(n\to \infty \right)$, presque-sûrement. Récemment, Devroye et Goudjil ont montré que si les $x$ sont les suites de Weyl, i.e., ${x}_{k}=\left\{\alpha k\right\},k=1,2,\cdots ,$$\alpha$ est une variable aléatoire uniforme sur [0,1], alors
 ${H}_{n}\left(x\right)\sim \left(12/{\pi }^{2}\right)lognloglogn,\phantom{\rule{1em}{0ex}}n\to \infty ,$
en probabilité. Dans ce papier nous considérons la classe de toutes les suites $x$ uniformément réparties pour lesquelles nous montrons que l’on a nécessairement ${H}_{n}\left(x\right)=o\left(n\right)$ quand $n\to \infty$.

@article{JTNB_2002__14_2_415_0,
author = {Michel Dekking and Peter Van der Wal},
title = {Uniform distribution modulo one and binary search trees},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {415--424},
publisher = {Universit\'e Bordeaux I},
volume = {14},
number = {2},
year = {2002},
zbl = {1075.11054},
mrnumber = {2040685},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_415_0/}
}
TY  - JOUR
AU  - Michel Dekking
AU  - Peter Van der Wal
TI  - Uniform distribution modulo one and binary search trees
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 415
EP  - 424
VL  - 14
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_415_0/
UR  - https://zbmath.org/?q=an%3A1075.11054
UR  - https://www.ams.org/mathscinet-getitem?mr=2040685
LA  - en
ID  - JTNB_2002__14_2_415_0
ER  - 
%0 Journal Article
%A Michel Dekking
%A Peter Van der Wal
%T Uniform distribution modulo one and binary search trees
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 415-424
%V 14
%N 2
%I Université Bordeaux I
%G en
%F JTNB_2002__14_2_415_0
Michel Dekking; Peter Van der Wal. Uniform distribution modulo one and binary search trees. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 415-424. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_415_0/

[1] M.V. Berry, J. Goldberg, Renormalisation of curlicues. Nonlinearity 1 (1988), 1-26. | MR | Zbl

[2] F.M. Dekking, S. De Graaf, L.E. Meester, On the node structure of binary search trees. In Mathematics and Computer Science - Algorithms, Trees, Combinatorics and Probabilities (Eds. Gardy, D. and Mokkadem, A.), pages 31-40. Birkhauser, Basel, 2000. | MR | Zbl

[3] F.M. Dekking, M. Mendès France, Uniform distribution modulo one: a geometrical viewpoint. J. Reine Angew. Math. 329 (1981), 143-153. | MR | Zbl

[4] J.-M. Deshouillers, Geometric aspect of Weyl sums. In Elementary and analytic theory of numbers (Warsaw, 1982), pages 75-82. PWN, Warsaw, 1985. | MR | Zbl

[5] L. Devroye, A note on the height of binary search trees. J. Assoc. Comput. Mach. 33 (1986), 489-498. | MR | Zbl

[6] L. Devroye, A. Goudjil, A study of random Weyl trees. Random Structures Algorithms 12 (1998), 271-295. | MR | Zbl

[7] C.A.R. Hoare, Quicksort. Comput. J. 5 (1962), 10-15. | MR | Zbl

[8] H.M. Mahmoud, Evolution of random search trees. John Wiley & Sons Inc., New York, 1992. Wiley-Interscience Series in Discrete Mathematics and Optimization. | MR | Zbl

[9] B. Pittel, Asymptotical growth of a class of random trees. Ann. Probab. 13 (1985), 414-427. | MR | Zbl