Patterns in Random Binary Search

In a randomly grown binary search tree (BST) of size n, any xed pattern occurs with a frequency that is on average proportional to n. Deviations from the average case are highly unlikely and well quantiied by a Gaussian law. Trees with forbidden patterns occur with an exponentially small probability that is characterized in terms of Bessel functions. The results obtained extend to BSTs a type of property otherwise known for strings and combinatorial tree models. They apply to paged trees or to quicksort with halting on short subbles. As a consequence, various pointer saving strategies for maintaining trees obeying the random BST model can be precisely quantiied. The methods used are based on analytic models, especially bivariate generating functions subjected to singularity perturbation asymptotics. Motifs dans les arbres binaires de recherche al eatoires R esum e. Dans un arbre binaire de recherche (ABR) de taille n construit par insertions al eatoires, chaque motif appara^ t avec une frequence qui est en moyenne proportionnelle a n. Les d eviations du cas moyen sont rares et bien quantii ees par une loi gaussienne. Les arbres a motifs exclus apparaissent avec une probabilit e exponentiellement petite caract eris ee en terme de fonctions de Bessel. Ces r esultats etendent aux ABR des propri et es connues par ailleurs dans le cas des cha^ nes de caract eres ou des arbres ob eissant aux mod eles com-binatoires uniformes. Ils s'appliquent a la pagination et aux arbres d'index ainsi qu'au comportement du \tri-rapide" (quicksort). Comme cons equence, plusieurs strat egies d'allocation de m emoire peuvent ^ etre pr ecis ement quantii ees. Les m ethodes utilis ees sont de nature analytique et reposent sur l'asymptotique de perturbation de singularit es appliqu ee aux s eries g en eratrices multivari ees. Abstract In a randomly grown binary search tree (BST) of size n, any xed pattern occurs with a frequency that is on average proportional to n. Deviations from the average case are highly unlikely and well quantiied by a Gaus-sian law. Trees with forbidden patterns occur with an exponentially small probability that is characterized in terms of Bessel functions. The results obtained extend to BSTs a type of property otherwise known for strings and combinatorial tree models. They apply to paged trees or to quicksort with halting on short subbles. As a consequence, various pointer saving strategies for maintaining trees obeying the random BST …