scholarly journals Analysis of some statistics for increasing tree families

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.

Red-black trees in a functional setting

1999 ◽  
Vol 9 (4) ◽  
pp. 471-477 ◽  
Author(s):  
CHRIS OKASAKI
Keyword(s):  
Computer Science ◽  
Binary Search ◽  
Binary Search Trees ◽  
Search Trees ◽  
Introductory Computer Science ◽  
Science Classes ◽  
Functional Setting

Everybody learns about balanced binary search trees in their introductory computer science classes, but even the stouthearted tremble at the thought of actually implementing such a beast. The details surrounding rebalancing are usually just too messy. To show that this need not be the case, we present an algorithm for insertion into red-black trees (Guibas and Sedgewick, 1978) that any competent programmer should be able to implement in fifteen minutes or less.

scholarly journals Numerical Studies of the Asymptotic Height Distribution in Binary Search Trees

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.

scholarly journals The Wiener Index of Random Trees

2002 ◽  
Vol 11 (6) ◽  
pp. 587-597 ◽  
Author(s):  
RALPH NEININGER
Keyword(s):  
Path Length ◽  
Probabilistic Models ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
Limit Laws ◽  
Recursive Trees ◽  
Asymptotic Correlations

The Wiener index is analysed for random recursive trees and random binary search trees in uniform probabilistic models. We obtain expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed point equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.

scholarly journals The height of q-Binary Search Trees

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.

scholarly journals Average depth in a binary search tree with repeated keys

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.

scholarly journals On the Diameter of Tree Associahedra

10.37236/7762 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Jean Cardinal ◽  
Stefan Langerman ◽  
Pablo Pérez-Lantero
Keyword(s):  
Computer Science ◽  
Search Tree ◽  
Binary Search ◽  
The Other ◽  
Discrete Mathematics ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Natural Notion ◽  
Minimum Number

We consider a natural notion of search trees on graphs, which we show is ubiquitous in various areas of discrete mathematics and computer science. Search trees on graphs can be modified by local operations called rotations, which generalize rotations in binary search trees. The rotation graph of search trees on a graph $G$ is the skeleton of a polytope called the graph associahedron of $G$.We consider the case where the graph $G$ is a tree. We construct a family of trees $G$ on $n$ vertices and pairs of search trees on $G$ such that the minimum number of rotations required to transform one search tree into the other is $\Omega (n\log n)$. This implies that the worst-case diameter of tree associahedra is $\Theta (n\log n)$, which answers a question from Thibault Manneville and Vincent Pilaud. The proof relies on a notion of projection of a search tree which may be of independent interest.

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