Binary Tree

Abstract. The restricted rotation distance dR(S, T) between two binary trees S, T of n vertices is the minimum number of rotations by which S can be transformed into T, where rotations can only take place at the root of the tree, or at the right child of the root. A sharp upper bound ...

Abstract-We describe a sequential universal data compression procedure for binary tree sources that performs the “double mixture.” Using a context tree, this method weights in an ef-ficient recursive way the coding distributions corresponding to all bounded memory tree sources, and ...

This paper develops the multidimensional binary search tree (or k -d tree, where k is the dimensionality of the search space) as a data structure for storage of information to be retrieved by associative searches. The k -d tree is defined and examples are given. It is shown to be quite efficient in its storage requirements. A significant advantage of this structure is that a single data structure can handle many types of queries very efficiently. Various utility algorithms are developed; their proven average running times in an n record file are: insertion, O (log n ); deletion of the root, O ( n ( k -1)/ k ); deletion of a random node, O (log n ); and optimization (guarantees logarithmic performance of searches), O ( n log n ). Search algorithms are given for partial match queries with t keys specified [proven maximum running time of O ( n ( k - t )/ k )] and for nearest neighbor queries [empirically observed average running time of O (log n ).] These performances far surpass the b...

We present an algorithm for the Merkle tree traversal problem which combines the efficient space-time trade-off from the fractal Merkle tree [3] and the space efficiency from the improved log space-time Merkle trees traversal [8]. We give an exhaustive analysis of the space and time efficiency of our algorithm in function of the parameters H (the height of the Merkle tree) and h (h = H L where L is the number of levels in the Merkle tree). We also analyze the space impact when a continuous deterministic pseudo-random number generator (PRNG) is used to generate the leaves. We further program a low storage-space and a low time-overhead version of the algorithm in Java and measure its performance with respect to the two different implementations cited above. Our implementation uses the least space when a continuous PRNG is used for the leaf calculation.

SHORT DESCRIPTION.  Demonstration that Cantor's diagonal argument is flawed and that real numbers, power set of natural numbers and power set of real numbers have the same cardinality as natural numbers.

ABSTRACT.  Cantor’s diagonal argument purports to prove that the set of real numbers is nondenumerably infinite.  I contend that the set of real numbers is, just like natural numbers, denumerably infinite.  In the first part of my paper, I show that Cantor’s diagonal argument is flawed, and point to the exact source and extent of its failure.  Then I use a similar criticism against another version of Cantor’s diagonalization maneuver, which he uses to prove that the power set of natural numbers is nondenumerably infinite.

In the second part of the paper, I propose an indirect method of establishing the denumerable infinity of real numbers (rather than directly finding a bijection between naturals and reals).  My method utilizes tree diagrams.  For convenience and simplicity, I employ a binary tree wherein the branches represent the real numbers in the interval [0, 1) written in binary symbolism.  Cantor’s diagonal argument concludes that the real numbers in the interval [0, 1) are nondenumerably infinite, and this suffices to establish that the entire set of real numbers are nondenumerably infinite.  I first take up the task of establishing that the real numbers in the interval [0, 1) are denumerably infinite, by way of showing that the segments of the binary tree (a segment is the line between any two consecutive nodes of the tree) can be paired off with natural numbers.  This way of “counting” the segments establishes that the set of segments is denumerably infinite.  I then prove that the set of branches of the tree, and thereby the reals in [0,1) they represent, cannot have a higher cardinality than the set of segments of the tree.  Hence the branches, and therefore the reals in [0, 1) represented by the braches, can only be denumerably infinite.  I then move on to show that not only the set of reals in [0, 1) but the entire set of reals is denumerably infinite.

Next I argue that two other contentions of Cantor’s are erroneous.  Against his claim that the power set of real numbers is nondenumerably infinite, with cardinality Aleph-2, I show that that set too is denumerably infinite, just as is the power set of natural numbers.  I end with some concluding remarks.

KEYWORDS.  Cantor’s diagonal argument, binary tree, cardinality of natural numbers, cardinality of real numbers, cardinality of the power set of natural numbers, cardinality of the power set of real numbers, Continuum Hypothesis

We herein introduce a new method of interpretable clustering that uses unsupervised binary trees. It is a three-stage procedure, the first stage of which entails a series of recursive binary splits to reduce the heterogeneity of the data within the new subsamples. During the second stage (pruning), consideration is given to whether adjacent nodes can be aggregated. Finally, during the third stage (joining), similar clusters are joined together, even if they do not share the same parent originally. Consistency results are obtained, and the procedure is used on simulated and real data sets.

There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the internal nodes of a binary tree by the numbers , indicating the sequence in which the nodes are visited. For given (size of the tree) and (a number between 1 and

In this paper, a new algorithm to find convex hull is presented which keeps the points into a form of nested binary tree. There are number of methods available for finding the convex hull but all of them are time consuming because they are based on comparing distances between the points and rotation of angles, whereas the proposed technique is based on binary tree which is efficient and more accurate than the existing algorithms as it decreases the computational cost and finds precise boundary of the object. Another main advantage of the proposed technique is the efficient memory usage because of binary tree structure.