Random Binary Trees Research Articles

On random cartesian trees

AbstractCartesian trees are binary search trees in which the nodes exhibit the heap property according to a second (priority) key. If the search key and the priority key are independent, and the trees is built based on n independent copies, Cartesian trees basically behave like ordinary random binary search trees. In this article, we analyze the expected behavior when the keys are dependent: in most cases, the expected search, insertion, and deletion times are Φ(√n). We indicate how these results can be used in the analysis of divide‐and‐conguer algorithms for maximal vectors and convex hulls. Finally, we look at distributions for which the expected time per operation grows like na for a ϵ[1/2, 1]. © 1994 John Wiley & Sons, Inc.

Horton-Strahler ordering of random binary trees

We study the Horton-Strahler ordering for random binary trees, which are statistically self-similar branching structures. Extending previously obtained results, we show that near the top of these trees, the expected bifurcation ratios tend strongly to the value 4. But at the root of the tree, the expected bifurcation ratio is less than 4, becoming asymptotically a periodic function of log4 n.

On the expected height of fringe-balanced trees

We study the effect of a well-known balancing heuristic on the expected height of a random binary search tree. After insertion of an element, if any node on the insertion path has a subtree of size precisely 2t+1 for a fixed integert, then the subtree rooted at that node is destroyed and replaced by a new subtree in which the median of the 2t+1 elements is the new root. IfH n denotes the height of the resulting random tree, we show thatH n /logn ?c(t) in probability for some functionc(t). In particular,c(0)=4.31107... (the ordinary binary search tree),c(1)=3.192570 ...,c(3)=2.555539 ...,c(10)=2.049289 ... andc(100)=1.623695 ....

An evaluation of self‐adjusting binary search tree techniques

AbstractMuch has been said in praise of self‐adjusting data structures, particularly self‐adjusting binary search trees. Self‐adjusting trees are most suited to skewed key‐access distributions as the techniques attempt to place the most commonly accessed keys near the root of the tree. Theoretical bounds on worst‐case and amortized performance (i.e. performance over a sequence of operations) have been derived which compare well with those for optimal binary search trees. In this paper, we compare the performance of three different techniques for self‐adjusting trees with that of AVL and random binary search trees. Comparisons are made for various tree sizes, levels of key‐access‐frequency skewness and ratios of insertions and deletions to searches. The results show that, because of the high cost of maintaining self‐adjusting trees, in almost all cases the AVL tree outperforms all the self‐adjusting trees and in many cases even a random binary search tree has better performance, in terms of CPU time, than any of the self‐adjusting trees. Self‐adjusting trees seem to perform best in a highly dynamic environment, contrary to intuition.

Counting and randomly generating binary trees

The number bn of binary trees with n nodes is well known. Two ways to determine bn: one is based on constructing and the other recognizes that bn is the some as the number of stack permutations. In this paper, a simple direct approach is used to find bn. It also provides another way to generate a random binary tree

The asymptotic contour process of a binary tree is a Brownian excursion

The contour process of a random binary tree t with n internal nodes is defined as the polygonal function constructed from the heights of the leaves of t (normalized by n ). We show that, as n → ∞, the limiting contour process is identical in distribution to a Brownian excursion.

The limiting common distribution of two leaf heights in a random binary tree

Soit B n la famille des arbres binaires a n sommets. On suppose que chaque arbre t∈B n est de meme probabilite. Nous determinons la distribution asymptotique commune des hauteurs des feuilles i et j (les feuilles etant enumerees de gauche a droite) quand i, j, n→∞, ainsi que i/n→x, j/n→y (0<x<y<1). Comme corollaire, la covariance asymptotique des deux hauteurs est determinee. Des applications sont indiquees

Generating binary trees at random

We give a new constructive proof of the Chung-Feller theorem. Our proof provides a new and simple linear-time algorithm for generating random binary trees on n nodes; the algorithm uses integers no larger than 2 n.

Limit laws for local counters in random binary search trees

AbstractLimit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for w‐dependent random variables. Examples include: the number of leaves (Ln), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply.

Applications of the theory of records in the study of random trees

The theory of records in sequences of independent identically distributed random variables leads to simple proofs of various properties of random trees, including among other things, the limit law for the depth of the last node of random ordered trees, random union-find trees, and random binary search trees.

Branching processes in the analysis of the heights of trees

It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random k—d trees, quadtrees and union-find trees under various models of randomization. For example, for the random binary search tree constructed from a random permutation of 1,..., n, it is shown that Hn/(c log(n)) tends to 1 in probability and in the mean as n→∞, where Hn is the height of the tree, and c =4.31107... is a solution of the equation \(\left( {\frac{{2e}}{c}} \right) = 1\). In addition, we show that \(H_n - c log (n) = {\text{O}}(\sqrt {log(n)loglog(n)} )\) in probability.

Parameter estimation in topological analysis of binary tree structures.

Different types of random binary topological trees (like neuronal processes and rivers) occur with relative frequencies that can be explained in terms of growth models. It will be shown how the model parameter determining the mode of growth can be estimated with the maximum likelihood procedure from observed data. Monte Carlo simulations were used to study the distributional properties of this estimator which appeared to have a negligible bias. It is shown that the minimum chi-square procedure yields an estimate that is very close to the maximum likelihood estimate. Moreover, the goodness-of-fit of the growth model can be inferred directly from the chi-square statistic. To illustrate the procedures we examined axonal trees from the goldfish tectum. A notion of complete partition randomness is presented as an alternative to our growth hypotheses.

The Expected Distribution of Degrees in Random Binary Search Trees

Etude de la distribution moyenne des degres des sommets dans un arbre de recherche binaire aleatoire

On growing random binary trees

A sequence { T n } n = 1 ∞ of nested binary trees generated by an infinite sequence of i.i.d. random variables is studied. Two absolute constants β 1, β 2 are shown to exist (0.37 < β 1 < 0.50, 3.58 < β 2 < 4.32), such that lim h n ln n = β 1 , limH n/ ln n = β 2 with probability one; here h n and H n are respectively the lengths of the shortest and the longest branches of the tree T n .

On Random Binary Trees

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 probability 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 asymptotically 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 all 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.

An empirical study of insertion and deletion in binary search trees

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.

Moments of particle size distributions under sequential breakage with applications to species abundance

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.

Moments of particle size distributions under sequential breakage with applications to species abundance

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.

A Naturally Occurring Function Continuous only at Irrationals

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

A principle of independence for binary tree searching

When searching unsuccessfully for a fixed element in a random binary search tree, the number of comparisons made whose result is "less" is independent from the number of comparisons whose result is "greater". This principle can be used to compute the mean and variance of the total number of comparisons involved in both a successful and an unsuccessful search.