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 $n\to \infty$.

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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},
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year = {2002},
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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/

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