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Scaling limit for the cover time of the λ-biased random walk on a binary tree with λ<1

David A. Croydon · Bernoulli · 2026

Cited by 0

Randomization of Search Trees by Subtree Size

C. Martínez, C. Mart Inez, Salvador Roura · 1995

In this paper we present probabilistic algorithms over random binary search trees such that: a) the insertion of a set of keys in any fixed order into an initially empty tree produces always a random tree; b) the deletion of any key of a random tree results in a random tree; c) the random choices made by the algorithms are based upon the sizes of the subtrees of the random tree, an information that can be used for rank searches, for instance; and d) the cost, measured as the number of visited nodes, of any elemental operation is the same as the cost of the standard deterministic version, with less than two expected rotation-like operations per update. 1. Introduction Given a binary search tree (BST, for short), common operations are the search of a given key in the tree and the retrieval of the information associated to that key if it is present, the insertion of a new item in the tree and the deletion of some item given its key. The symmetric or inorder traversal of any binar...

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Randomized binary search trees

Martínez Parra, Conrado, Roura Ferret, Salvador · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 1997

In this paper we present randomized algorithms over binary search\n\t\t\t\t trees such that: a) the insertion of a set of keys, in any fixed\n\t\t\t\t order, into an initially empty tree always produces a random binary\n\t\t\t\t search tree; b) the deletion of any key from a random binary search\n\t\t\t\t tree results in a random binary search tree; c) the random choices\n\t\t\t\t made by the algorithms are based upon the sizes of the subtrees of the\n\t\t\t\t tree; this implies that we can support accesses by rank without\n\t\t\t\t additional storage requirements or modification of the data\n\t\t\t\t structures; and d) the cost of any elementary operation, measured as\n\t\t\t\t the number of visited nodes, is the same as the expected cost of its\n\t\t\t\t standard deterministic counterpart; hence, all search and update\n\t\t\t\t operations have guaranteed expected cost O(log n), but\n\t\t\t\t now irrespective of any assumption on the input distribution.

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طريقة لإنشاء الأشجار الثنائية بشكل عشوائي

Saud M.A. [سعود محمد عبد الله مغربي] Maghrabi · Qatar University QSpace (Qatar University) · 2000

There are two widely used random distributions on binary trees, namely the binary search tree distribution and the uniform distribution. By analyzing the performance of algorithms that manipulate binary trees generated at random, it is possible to assess the average case performance of the algorithm. This average case performance is often a more useful reflection of the algorithm's suitability than the worst-case performance. An algorithm may have a worst-case run time that is of exponential time complexity, but an average-case run time that is of polynomial time complexity. In such cases, one wishes to assess the performance of an algorithm on a 'typical' range of tree structures and thus relate properties of randomly generated trees, such as depth, to aspects affecting the algorithm's run-time. The purpose of this paper is to design and implement a random binary trees generation algorithm that considers the average case performance. The algorithm is differ from both the uniform distribution and the binary search tree distribution. Analysis of the behaviour of the algorithm is given.

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Analysis of ${\textit{d}}$-ary tree algorithms with successive interference cancellation

Quirin Vogel, Yash Deshpande, Čedomir Stefanović et al. · Journal of Applied Probability · 2024

Abstract We calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximizes throughput. Our method works with many observables and can be used as a blueprint for further analysis.

Cited by 2Open access

Primed in t, S..4 ON RANDOM BINARY TREES*

Brown Gerald G, Gerald G. Brown, Bruno et al. · 1984

A widely used class of binary trees is studied in order to provide information useful in evaluating algorithms based on this storage structure. A closed form counting formula for the number of binary trees with n nodes and height k is developed and restated as a recursion more useful computationally. A generating function for the number of nodes given height is developed and used to find the asymptotic distribution of binary trees. An asymptotic probab ility distribution for height given the number of nodes is derived based on equally likely binary trees. This is compared with a similar result for general trees. Random binary trees (those resulting from a binary tree sorting algorithm applied to random strings of symbols) are counted in terms of the mapping of permutations of n symbols to binary trees of height k. An explicit formula for this number is given with an equivalent recursive definition for computational use. A generating function is derived for the number of symbols given height. Lower and upper bounds on random binary tree height are developed and shown to approach one another asymptot ically as a function of n, providing a limiting expression for the expected height. The random binary trees are examined further to provide expressions for the expectations of the number of vacancies at each level, the distribution of vacancies over alI levels. the comparisons required for insertion of a new random symbol, the fraction of nodes occupied at a particular level. the number of leaves, the number of single vacancies at each level, and the number of twin vacancies at each level. A random process is defined for the number of symbols required to grow a tree exceeding any given height. Finally. an appendix is given with sample tabulations and figures of the distributions.

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Oblique splits in artificial representative trees for random forests

Laabs, Björn-Hergen, Bakare, Janet, Szymczak, Silke · German Medical Science (German Research Foundation) · 2025

Introduction: Artificial representative trees (ARTs) are an interpretation method for random forests where a single binary decision tree is generated as a surrogate model to be interpreted instead of the whole ensemble [ref:1]. While random forests, as an ensemble of binary decision trees, [for full text, please go to the a.m. URL]

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An empirical study of insertion and deletion in binary search trees

Jeffrey L. Eppinger · Communications of the ACM · 1983

This paper describes an experiment on the effect of insertions and deletions on the path length of unbalanced binary search trees. Repeatedly inserting and deleting nodes in a random binary tree yields a tree that is no longer random. The expected internal path length differs when different deletion algorithms are used. Previous empirical studies indicated that expected internal path length tends to decrease after repeated insertions and asymmetric deletions. This study shows that performing a larger number of insertions and asymmetric deletions actually increases the expected internal path length, and that for sufficiently large trees, the expected internal path length becomes worse than that of a random tree. With a symmetric deletion algorithm, however, the experiments indicate that performing a large number of insertions and deletions decreases the expected internal path length, and that the expected internal path length remains better than that of a random tree.

Cited by 24Open access

Inducibility in binary trees and crossings in random tanglegrams

Éva Czabarka, Łászló A. Székely, Stephan M. Wagner · arXiv (Cornell University) · 2016

In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves, we let $\gamma(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that induce a tree isomorphic to $B$. The inducibility of $B$ is $\limsup_{|T| \to \infty} \gamma(B,T)$. We determine the inducibility in some special cases, show that every binary tree has positive inducibility and prove that caterpillars are the only binary trees with inducibility $1$. We also formulate some open problems and conjectures on the inducibility. Finally, we present an application to crossing numbers of random tanglegrams.

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Doob-Martin boundary of Rémy's tree growth chain

Evans, SN, Grübel, R, Wakolbinger, A · eScholarship (California Digital Library) · 2017

Rémy's algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the nth tree is uniformly distributed over the set of rooted, planar, binary trees with 2n + 1 vertices. We obtain a concrete characterization of the Doob-Martin boundary of this transient Markov chain and thereby delineate all the ways in which, loosely speaking, this process can be conditioned to "go to infinity" at large times. A (deterministic) sequence of finite rooted, planar, binary trees converges to a point in the boundary if for each m the random rooted, planar, binary tree spanned by m + 1 leaves chosen uniformly at random from the nth tree in the sequence converges in distribution as n tends to infinity-a notion of convergence that is analogous to one that appears in the recently developed theory of graph limits. We show that a point in the Doob-Martin boundary may be identified with the following ensemble of objects: a complete separable ℝ-tree that is rooted and binary in a suitable sense, a diffuse probability measure on the R-tree that allows us to make sense of sampling points from it, and a kernel on the R-tree that describes the probability that the first of a given pair of points is below and to the left of their most recent common ancestor while the second is below and to the right. Two such ensembles represent the same point in the boundary if for each m the random, rooted, planar, binary trees spanned by m + 1 independent points chosen according to the respective probability measures have the same distribution. Also, the Doob-Martin boundary corresponds bijectively to the set of extreme point of the closed convex set of nonnegative harmonic functions that take the value 1 at the binary tree with 3 vertices; in other words, the minimal and full Doob-Martin boundaries coincide. These results are in the spirit of the identification of graphons as limit objects in the theory of graph limits.

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Analysis of algorithms on threaded trees

Keith Brinck · The Computer Journal · 1981

This paper investigates the performances of various algorithms operating on binary trees. In particular the expected times necessary to compute the different successor and predecessor nodes of a random node in a random threaded binary tree are presented. Based on these results it is then shown that the inorder traversal algorithm using threads is about 25% better performing than the corresponding algorithm using stacks, while for preorder and postorder little or no improvement in traversal times is gained by the use of threads. Finally the presence of threads in a binary tree is briefly discussed with respect to the performance of insertion and deletion algorithms.

Cited by 19

A note on the independence number, domination number and related\n parameters of random binary search trees and random recursive trees

Michael Fuchs, Cecilia Holmgren, Dieter Mitsche et al. · arXiv (Cornell University) · 2019

We identify the mean growth of the independence number of random binary\nsearch trees and random recursive trees and show normal fluctuations around\ntheir means. Similarly we also show normal limit laws for the domination number\nand variations of it for these two cases of random tree models. Our results are\nan application of a recent general theorem of Holmgren and Janson on fringe\ntrees in these two random tree models.\n

Cited by 6Open access

The height of record‐biased trees

Benoît Corsini · Random Structures and Algorithms · 2022

Abstract Given a permutation , its corresponding binary search tree is obtained by recursively inserting the values into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In this article, we study the height of binary search trees drawn from the record‐biased model of permutations whose probability measure on the set of permutations is proportional to , where . We show that the height of a binary search tree built from a record‐biased permutation of size with parameter is of order , hence extending previous results of Devroye on the height or random binary search trees.

Cited by 0Open access

Value

Depends R, Bjoern Menze, Nico Splitthoff et al. · 2014

Description Random forest with oblique decision trees for binary classification tasks. Discriminative node models in the tree are based on: ridge regression, partial least squares regression, logistic regression, linear support vector machines, or random coefficients.

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Value

Depends R, Bjoern Menze, Nico Splitthoff et al. · 2013

Description Random forest with oblique decision trees for binary classification tasks. Discriminative node models in the tree are based on: ridge regression, partial least squares regression, logistic regression, linear support vector machines, or random coefficients.

Cited by 0

A Logarithmic Random Access Tree

Tom Morgan · 2007

Making a data structure that performs like a dynamic array but functions in logarithmic time for all operations is the goal, but is by no means a trivial one. The obvious solution is to use a tree of some sort, but how? By using a binary tree in which values are stored at the leaf nodes and each node keeps track of how many leaves there are below it, we can quickly achieve logarithmic random access, insertion and deletion in the average case but all operations are linear on the worst case. To balance the tree, a self-balancing binary search tree like a Red Black Tree or a Splay Tree is used. These trees are always balanced and by using their balancing mechanisms and changing them to use the random access system described above, the goal is reached.

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Generating Random Binary Trees - A Survey

Trees A Survey, Erkki Mäkinen · 1998

This paper surveys algorithms for generating unbiased random binary trees. There exist several linear time algorithms. The best algorithms use only integers of size O(n) to generate binary trees on n nodes. 1 Introduction Binary trees are essential in various branches of computer science [7]. In time to time, there is a need to generate random binary trees. For example, when writing a program to manipulate binary trees, it is advantageous to have an efficient method to genarate random binary trees with some fixed number n of nodes in order to test the program. Mathematically, there are no problems at all in generating random binary trees. Namely, there exist algorithms to enumerate binary trees. Simply choose a random natural number i from the interval [1::B n ], where the nth Catalan number B n = / 2n n ! 1 n + 1 (1) gives the number of binary trees with n nodes, and output the ith binary tree from the enumeration. Computationally the problem is more complicated. As Martin a...

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A Naturally Occurring Function Continuous only at Irrationals

A. Bagchi, Edward M. Reingold · American Mathematical Monthly · 1982

computer science. The function f arises in connection with random binary search trees. We shall only sketch the basic definitions and relevant background material, referring the reader to [2] or [3] for a comprehensive presentation. A binary tree is a finite set of nodes that is either empty or consists of a root and two disjoint binary trees called the left and right subtrees of the root. It is customary to draw such a tree as

Cited by 4

Search trees: Metric aspects and strong limit theorems

Grübel, Rudolf · arXiv (Cornell University) · 2012

We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces converge with probability 1. This is then used to obtain almost sure convergence for various tree functionals, together with representations of the respective limit random variables as functions of the limit tree.

Cited by 0Open access

Search trees: metric aspects and strong limit theorems

Rudolf Grübel, Leibniz Universität Hannover · 2012

We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces converge with probability 1. This is then used to obtain almost sure convergence for various tree functionals, together with representations of the respective limit random variables as functions of the limit tree.

Cited by 10