Numerical Studies of the Asymptotic Height Distribution in Binary Search Trees

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.319 ◽

2003 ◽

Vol Vol. 6 no. 1 ◽

Author(s):

Charles Knessl

Keyword(s):

Integral Equation ◽

Binary Search ◽

Linear Integral Equation ◽

Height Distribution ◽

Binary Search Trees ◽

Search Trees ◽

Asymptotic Results ◽

Numerical Studies ◽

International Audience ◽

Linear Integral

International audience We study numerically a non-linear integral equation that arises in the study of binary search trees. If the tree is constructed from n elements, this integral equation describes the asymptotic (as n→∞) distribution of the height of the tree. This supplements some asymptotic results we recently obtained for the tails of the distribution. The asymptotic height distribution is shown to be unimodal with highly asymmetric tails.

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Analysis of some statistics for increasing tree families

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.326 ◽

2004 ◽

Vol Vol. 6 no. 2 ◽

Author(s):

Alois Panholzer ◽

Helmut Prodinger

Keyword(s):

Computer Science ◽

Normal Distribution ◽

Probabilistic Models ◽

Binary Search ◽

Rooted Trees ◽

Binary Search Trees ◽

Search Trees ◽

Ordered Trees ◽

International Audience

International audience This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many mportant tree families that have applications in computer science or are used as probabilistic models in various applications, like \emphrecursive trees, heap-ordered trees or \emphbinary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters \textitdepth of a randomly chosen node, \textitdistance between two randomly chosen nodes, and the generalisations where \textitp nodes are randomly chosen Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures.

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The height of q-Binary Search Trees

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.304 ◽

2002 ◽

Vol Vol. 5 ◽

Author(s):

Michael Drmota ◽

Helmut Prodinger

Keyword(s):

Geometric Distribution ◽

Binary Search ◽

Binary Search Trees ◽

Search Trees ◽

International Audience

International audience q-binary search trees are obtained from words, equipped with a geometric distribution instead of permutations. The average and variance of the heighth computated, based on random words of length n, as well as a Gaussian limit law.

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Average depth in a binary search tree with repeated keys

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3496 ◽

2006 ◽

Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽

Author(s):

Margaret Archibald ◽

Julien Clément

Keyword(s):

Search Tree ◽

Binary Search ◽

Binary Search Tree ◽

Average Depth ◽

Binary Search Trees ◽

Search Trees ◽

Random Sequences ◽

International Audience ◽

The Right

International audience Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.

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