Class Of Binary Trees Research Articles

Counting smaller elements in the Tamari and m-Tamari lattices

We introduce new combinatorial objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear operator that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear operator on each tree T counts the number of trees smaller than T in the Tamari order.Then we show that a similar (m+1)-linear operator is also used in the functional equation of m-Tamari intervals. We explain how the m-Tamari lattices can be interpreted in terms of (m+1)-ary trees or a certain class of binary trees. We then use the interval-posets to recover the functional equation of m-Tamari intervals and to prove a generalized formula that counts the number of elements smaller than or equal to a given tree in the m-Tamari lattice.

A note on the cubical dimension of new classes of binary trees

The cubical dimension of a graph G is the smallest dimension of a hypercube into which G is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with 2n vertices, n ⩾ 1, is n. The 2-rooted complete binary tree of depth n is obtained from two copies of the complete binary tree of depth n by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.

Grid Drawings of Binary Trees: An Experimental Study

In this paper we consider the class of binary trees and present the results of a comprehensive experimental study on the four most representative algorithms for drawing trees, one for each of the following treedrawing approaches: Separation-Based, Path-based, Level-based, and Ringed Circular Layout. We establish a simpler, more intuitive format for storing binary trees in files and create a large suite of randomlygenerated, unbalanced, complete, AVL, Fibonacci, and molecular combinatory binary trees of various sizes. Our study is therefore conducted on randomly-generated, unbalanced, and AVL binary trees with between 100 and 50, 000 nodes, on Fibonacci trees Tn for n = 1, 2, ..., 45, 46 (143 to 46, 367 nodes), on complete binary trees of size 2−1 for n = 7, 8, ..., 15, 16 (127 to 65, 535 nodes), and on molecular combinatory binary trees with between 133 and 50, 005 nodes. Our study yields 70 charts comparing the performance of the drawing algorithms with respect to ten quality measures, namely Area, Aspect Ratio, Size, Total Edge Length, Average Edge Length, Maximum Edge Length, Uniform Edge Length, Angular Resolution, Closest Leaf, and Farthest Leaf. None of the algorithms has been found to be the best in all categories. This observation leads us to create an adaptive system that determines the type of a binary tree and then selects an algorithm to draw the tree depending upon the specified quality measures. Currently, our adaptive tree drawing system recognizes all six types of binary trees and all ten measures included in our experimental study. Under our settings, our adaptive tree drawing system outperforms any system using a single binary tree drawing algorithm. Submitted: April 2007 Reviewed: June 2007 Revised: August 2007 Accepted: October 2007 Final: November 2007 Published: June 2008 Article type: Regular Paper Communicated by: S. Kobourov E-mail addresses: rusu@rowan.edu (Adrian Rusu) santia09@students.rowan.edu (Confesor Santiago) 132 Rusu & Santiago Grid Drawings of Binary Trees

The alternation hierarchy in fixpoint logic with chop is strict too

Fixpoint Logic with Chop extends the modal μ -calculus with a sequential composition operator which results in an increase in expressive power. We develop a game-theoretic characterisation of its model checking problem and use these games to show that the alternation hierarchy in this logic is strict. The structure of this result follows the lines of Arnold’s proof showing that the alternation hierarchy in the modal μ -calculus is strict over the class of binary trees.

Scheduling trees with large communication delays on two identical processors

We consider the problem of scheduling trees on two identical processors in order to minimize the makespan. We assume that tasks have unit execution times, and arcs are associated with large identical integer communication delays. We prove that the problem is NP-hard in the strong sense even when restricted to the class of binary trees, and we provide a polynomial-time algorithm for complete binary trees.

On embedding binary trees into hypercubes

It is conjectured that an N-vertex binary tree can be embedded into a ⌈log N⌉-dimensional cube with a dilation of at most 2. Although it is known that an N-vertex binary tree can be embedded into a ⌈log N⌉-dimensional cube with a dilation of at most 8, the conjecture has been verified only for some restricted classes of binary trees. This paper verifies the conjecture for wider classes of binary trees. That is, we show that a 2n-vertex balanced one-legged caterpillar with leg length of at most 2 can be embedded into an n-dimensional cube with dilation 1, and that an N-vertex binary tree with proper pathwidth of at most 2 can be embedded into a ⌈log N⌉-dimensional cube with dilation 2. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 83(1): 1–13, 2000

Growing binary trees in a random environment

A class of binary trees that grow in a random environment, where the state of the environment can change at every vertex of the trees, is studied. The trees considered are single-type and two-type binary trees that grow in a two-state Markovian environment. For each kind of tree, the conditions on the environment process for extinction of the tree are determined, and the problem of calculating the expected number of vertices of the tree is addressed. Different ways of growing the trees are compared. >

Generalized binary split trees

The definition of binary split trees is generalized by removing the condition of decreasing frequency. It is shown that the access time of generalized split trees is less than that of split trees in general. The optimal representation of generalized split trees is studied. A polynomial time algorithm to construct such optimal tree structures is given. The relationship among several classes of binary trees are also discussed.

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.

Symmetric binary B-Trees: Data structure and maintenance algorithms

A class of binary trees is described for maintaining ordered sets of data. Random insertions, deletions, and retrievals of keys can be done in time proportional to log N where N is the cardinality of the data-set. Symmetric B-Trees are a modification of B-trees described previously by Bayer and McCreight. This class of trees properly contains the balanced trees.