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This article introduces the basic concepts of binary trees, and then works through a series of practice problems with solution code in C/C++ and Java. Binary trees have an elegant recursive pointer structure, so they are a good way to learn recursive pointer algorithms.
Computer Science Education
Background and Context: Recursion in binary trees has proven to be a hard topic. There was not much research on enhancing student understanding of this topic. Objective: We present a tutorial to enhance learning through practice of recursive operations in binary trees, as it is typically taught post-CS2. Method: We identified the misconceptions students have in recursive operations on binary trees. We designed a code writing exam question to measure those misconceptions. We built a tutorial that trains students on avoiding those misconceptions through the use of a semantic code analyzer that detects misconceptions and provides appropriate feedback. Findings: Our results show an improvement in student performance when using the tutorial along with the practice exercises, and even more improvement when the same exercises are used with a semantic code analyzer. Implications: The best way to use our tutorial to enhance student performance on advanced recursion is to allow students solving the tutorial exercises with the the semantic feedback.
International Journal of Advanced Computer Science and Applications, 2015
Study of binary trees has prominent place in the training course of DSA (Data Structures and Algorithms). Their implementation in C++ however is traditionally difficult for students. To a large extent these difficulties are due not so much to the complexity of algorithms as to language complexity in terms of memory management by raw pointersthe programmer must consider too many details to ensure a reliable, efficient and secure implementation. Evolution of C++ regarded to automated resource management, as well as experience in implementation of linear lists by means of C++ 11/14 lead to an attempt to implement binary search trees (BST) via smart pointers as well. In the present paper, the authors share experience in this direction. Some conclusions about pedagogical aspects and effectiveness of the new classes, compared to traditional library containers and implementation with built-in pointers, are made.
ACM SIGCSE Bulletin, 1996
The Binary Search Tree serves as an important example when teaching data structures. We explore new approaches to understanding the implementation of a Binary Search Tree, using concepts from Object-Oriented Programming and C++. The Binary Search Tree illustrates how adopting a new approach and a new language can lead to a new way of thinking about a familiar problem.
Trees: Basic tree concepts, Binary Trees: Properties, Representation of Binary Trees using arrays and linked lists, operations on a Binary tree , Binary Tree Traversals (recursive), Creation of binary tree from in, pre and post order traversals A tree is hierarchical collection of nodes. One of the nodes, known as the root, is at the top of the hierarchy. Each node can have at most one link coming into it. The node where the link originates is called the parent node. The root node has no parent. The links leaving a node (any number of links are allowed) point to child nodes. Trees are recursive structures. Each child node is itself the root of a subtree. At the bottom of the tree are leaf nodes, which have no children.
The modeling of dynamical systems from a time series implemented by our DSA program introduces binary trees of height with all leaves on the same level, and the related subtrees of height L <= D. These are called epsilon-trees and epsilon-subtrees. The recursive and nonrecursive versions of the traversal algorithms for the trees with dynamically created nodes are discussed. The original nonrecursive algorithms that return the pointer to the next node in preorder, inorder and postorder traversals are presented. The space-time complexity analysis shows, and the execution time measurements confirm, that for these algorithms the recursive versions have approximately 10-25% better time constants. Still, the use of nonrecursive algorithms may be more appropriate in several occasions.
ARTICLE INFO Binary trees are essential structures in Computer Science. The leaf (leaves) of a binary tree is one of the most significant aspects of it. In this study, we prove that the order of a leaf (leaves) of a binary tree is the same in the main tree traversals; preorder, inorder, and postorder. Then, we prove that given the preorder and postorder traversals of a binary tree, the leaf (leaves) of a binary tree can be determined. We present the algorithm BT-leaf, a novel one, to detect the leaf (leaves) of a binary tree from its preorder and postorder traversals in quadratic time and linear space.
2018
declare that this thesis "Analysis of Tree in Computer-based Application", Thesis Title and the work presented in it are our own. We confirm that: This work was done wholly or mainly while in candidature for a BSc degree at United International University. Where any part of this thesis has previously been submitted for a degree or any other qualification at United International University or any other institution, this has been clearly stated. Where we have consulted the published work of others, this is always clearly attributed. Where we have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely our own work. We have acknowledged all main sources of help. Where the thesis is based on work done by ourselves jointly with others, we have made clear exactly what was done by others and what we have contributed ourselves.
Journal of Algorithms and Computation, 2017
Binary trees are essential structures in Computer Science. The leaf (leaves) of a binary tree is one of the most significant aspects of it. In this study, we prove that the order of a leaf (leaves) of a binary tree is the same in the main tree traversals; preorder, inorder, and postorder. Then, we prove that given the preorder and postorder traversals of a binary tree, the leaf (leaves) of a binary tree can be determined. We present the algorithm BT-leaf, a novel one, to detect the leaf (leaves) of a binary tree from its preorder and postorder traversals in quadratic time and linear space. http://jac.ut.ac.ir/article_406_49.html
The Computer Journal, 1998
For the analysis of an algorithm to generate binary trees, the behaviour of a certain sequence of numbers is essential. In the original paper, it was expressed by a recursion. Here, we show how to solve this (and similar) recursions, both explicitly and asymptotically. Some additional information about useful mathematical software is also provided.
Proceedings of the ACM SIGPLAN 1992 conference on Programming language design and implementation - PLDI '92, 1992
Even though impressive progress has been made in the area of optimizing and parallelizing programs with arrays, the application of similar techniques to programs with pointer data structures has remained di cult. In this paper we introduce a new approach that leads to improved analysis and transformation of programs with recursively-de ned pointer data structures. Our approach is based on a mechanism for the Abstract Description of Data Structures (ADDS), which makes explicit the important properties, such as dimensionality, of pointer data structures. Numerous examples demonstrate that ADDS de nitions are both natural to specify and exible enough to describe complex, cyclic pointer data structures. We discuss how an abstract data structure description can improve program analysis by presenting an analysis approach that combines an alias analysis technique, path matrix analysis, with information available from an ADDS declaration. Given this improved alias analysis technique, we provide a concrete example of applying a software pipelining transformation to loops involving pointer data structures.
1974
This project documents the results of an investigation into binary search trees. Because of their favourable characteristics binary search trees have become popular for information storage and retrieval applications in a one level store. The trees may be of two types, weighted and unweighted. Various algorithms are presented, in a machine independent context, for both types and an empirical evaluation is performed. An important software aid used for graphically displaying a binary tree is also described.
2012 International Conference on Information Technology Based Higher Education and Training (ITHET), 2012
The computer science students mostly face with the difficulties in learning the topics of algorithms courses. Only listening the topic from the teacher or just writing makes the learning volatile. Instead of listening or writing, if there is something visual, it would be more permanent to learn because visuality increases the learning potential and the time for learning is minimized. The adversities of classical education techniques were intended to be eliminated in this study via computer games which are becoming more and more popular in this age. An educational convenience is provided for the subject of tree traversal algorithms. Tree traversal algorithms are one of the basic and confused concepts in algorithms and programming courses in computer science. A game called "binary apple tree" was established to teach and learn the subject easier.
2007
Preface functions and complete programs of substantial length. The exercises and programming projects, moreover, constitute an indispensable part of the book. Many of these are immediate applications of the topic under study, often requesting that programs be written and run, so that algorithms may be tested and compared. Some are larger projects, and a few are suitable for use by a small group of students working together. Our programs are written in the popular object-oriented language C++. We take the view that many object-oriented techniques provide natural implementations for basic principles of data-structure design. In this way, C++ allows us to construct safe, efficient, and simple implementations of data-structures. We recognize that C++ is sufficiently complex that students will need to use the experience of a data structures courses to develop and refine their understanding of the language. We strive to support this development by carefully introducing and explaining various object-oriented features of C++ as we progress through the book. Thus, we begin Chapter 1 assuming that the reader is comfortable with the elementary parts of C++ (essentially, with the C subset), and gradually we add in such object-oriented elements of C++ as classes, methods, constructors, inheritance, dynamic memory management, destructors, copy constructors, overloaded functions and operations, templates, virtual functions, and the STL. Of course, our primary focus is on the data structures themselves, and therefore students with relatively little familiarity with C++ will need to supplement this text with a C++ programming text. SYNOPSIS By working through the first large project (CONWAY's game of Life), Chapter 1 Programming Principles expounds principles of object-oriented program design, top-down refinement, review, and testing, principles that the student will see demonstrated and is expected to follow throughout the sequel. At the same time, this project provides an opportunity for the student to review the syntax of elementary features of C++, the programming language used throughout the book. Chapter 2 introduces the first data structure we study, the stack. The chapter Preface • Synopsis xiii destructors, copy constructors, and overloaded assignment operators, all of which are needed in the safe C++ implementation of linked structures. Chapter 5 continues to elucidate stacks by studying their relationship to prob-Recursion lem solving and programming with recursion. These ideas are reinforced by exploring several substantial applications of recursion, including backtracking and tree-structured programs. This chapter can, if desired, be studied earlier in a course than its placement in the book, at any time after the completion of Chapter 2. More general lists with their linked and contiguous implementations provide Lists and Strings the theme for Chapter 6. The chapter also includes an encapsulated string implementation, an introduction to C++ templates, and an introduction to algorithm analysis in a very informal way. Chapter 7, Chapter 8, and Chapter 9 present algorithms for searching, sorting, Searching and table access (including hashing), respectively. These chapters illustrate the interplay between algorithms and the associated abstract data types, data struc-Sorting tures, and implementations. The text introduces the "big-O " and related notations for elementary algorithm analysis and highlights the crucial choices to be made regarding best use of space, time, and programming effort. These choices require Tables and Information Retrieval that we find analytical methods to assess algorithms, and producing such analyses is a battle for which combinatorial mathematics must provide the arsenal. At an elementary level we can expect students neither to be well armed nor to possess the mathematical maturity needed to hone their skills to perfection. Our goal, therefore, is to help students recognize the importance of such skills in anticipation of later chances to study mathematics. Binary trees are surely among the most elegant and useful of data structures. Their study, which occupies Chapter 10, ties together concepts from lists, searching, Binary Trees and sorting. As recursively defined data structures, binary trees afford an excellent opportunity for the student to become comfortable with recursion applied both to data structures and algorithms. The chapter begins with elementary topics and progresses as far as such advanced topics as splay trees and amortized algorithm analysis. Chapter 11 continues the study of more sophisticated data structures, including Multiway Trees tries, B-trees, and red-black trees. Chapter 12 introduces graphs as more general structures useful for problem Graphs solving, and introduces some of the classical algorithms for shortest paths and minimal spanning trees in graphs. The case study in Chapter 13 examines the Polish notation in considerable detail, exploring the interplay of recursion, trees, and stacks as vehicles for problem Case Study: The Polish Notation solving and algorithm development. Some of the questions addressed can serve as an informal introduction to compiler design. As usual, the algorithms are fully developed within a functioning C++ program. This program accepts as input an expression in ordinary (infix) form, translates the expression into postfix form, and evaluates the expression for specified values of the variable(s). Chapter 13 may be studied anytime after the completion of Section 10.1. The appendices discuss several topics that are not properly part of the book's subject but that are often missing from the student's preparation. Appendix A presents several topics from discrete mathematics. Its final two Mathematical Methods sections, Fibonacci numbers amd Catalan numbers, are more advanced and not xiv Preface needed for any vital purpose in the text, but are included to encourage combinatorial interest in the more mathematically inclined. Appendix B discusses pseudorandom numbers, generators, and applications, Random Numbers a topic that many students find interesting, but which often does not fit anywhere in the curriculum. Appendix C catalogues the various utility and data-structure packages that are Packages and Utility Functions developed and used many times throughout this book. Appendix C discusses declaration and definition files, translation units, the utility package used throughout the book, and a package for calculating CPU times. Appendix D, finally, collects all the Programming Precepts and all the Pointers Programming Precepts, Pointers, and Pitfalls and Pitfalls scattered through the book and organizes them by subject for convenience of reference. COURSE STRUCTURE The prerequisite for this book is a first course in programming, with experience prerequisite using the elementary features of C++. However, since we are careful to introduce sophisticated C++ techniques only gradually, we believe that, used in conjunction with a supplementary C++ textbook and extra instruction and emphasis on C++ language issues, this text provides a data structures course in C++ that remains suitable even for students whose programming background is in another language such as C, Pascal, or Java. A good knowledge of high school mathematics will suffice for almost all the algorithm analyses, but further (perhaps concurrent) preparation in discrete mathematics will prove valuable. Appendix A reviews all required mathematics. This book is intended for courses such as the ACM Course CS2 (Program Design content and Implementation), ACM Course CS7 (Data Structures and Algorithm Analysis), or a course combining these. Thorough coverage is given to most of the ACM/IEEE knowledge units 1 on data structures and algorithms. These include: AL1 Basic data structures, such as arrays, tables, stacks, queues, trees, and graphs; AL2 Abstract data types; AL3 Recursion and recursive algorithms; AL4 Complexity analysis using the big Oh notation; AL6 Sorting and searching; and AL8 Practical problem-solving strategies, with large case studies. The three most advanced knowledge units, AL5 (complexity classes, NP-complete problems), AL7 (computability and undecidability), and AL9 (parallel and distributed algorithms) are not treated in this book.
It is well-known that, given inorder traversal along with one of the preorder or postorder traversals of a binary tree, the tree can be determined uniquely. Several algorithms have been proposed to reconstruct a tree from its inorder and preorder traversals as well as inorder and postorder traversals. There is no study to focus on reconstructing a binary tree from both its preorder and postorder traversals. In this paper, we proved that given preorder and postorder traversals of a binary tree, the tree may not be identified uniquely, however, determining all the feasible solution(s) is possible. We present the PrePos algorithm, a novel algorithm to reconstruct all the possible binary tree(s) from its preorder and postorder traversals. PrePos algorithm not only finds the all the possible solutions, but also determines different types of the nodes in a binary tree; nodes with two children, nodes with one child, and node with no child. In the end, PrePos returns a matrix-based structure to represent all the binary tree solution(s). By this representation, the number of feasible solution can be counted in linear time.
ACM SIGCSE Bulletin, 1996
By applying object-oriented design to the definition of a binary search tree, Berman and Duvall [1] designed a data structure comprised of three classes: (i) an EmptyBST class to model empty binary search trees, (ii) a NonEmptyBST class to model non-empty binary search trees, and (iii) a BST base class for common attributes of EmptyBST and NonEmptyBST objects. That paper noted the problem of inserting new values into such a structure: since insertions occur at an EmptyBST object, an EmptyBST would have to "turn into" a NonEmptyBST; a behavior beyond the capabilities of the classes in most languages.
2008
By applying object-oriented design to the definition of a binary search tree, Berman and Duvall [1] designed a data structure comprised of three classes: (i) an EmptyBST class to model empty binary search trees, (ii) a NonEmptyBST class to model non-empty binary search trees, and (iii) a BST base class for common attributes of EmptyBST and NonEmptyBST objects. That paper noted the problem of inserting new values into such a structure: since insertions occur at an EmptyBST object, an EmptyBST would have to "turn into " a NonEmptyBST; a behavior beyond the capabilities of the classes in most languages. This paper presents three C++ solutions to the insertion problem in their order of development. The first solution uses a procedural programming technique, with the second and third solutions shifting to a more object-oriented approach. This chronology illustrates the author's ongoing battle to shift from procedural to object-oriented thinking.
2012
Binary tree traversal refers to the process of visiting each node in a specified order. Given the inorder traversal of a binary tree, along with one of its preorder or postorder traversals, the original binary tree can be uniquely identified. Many recursive and non recursive method of construction of the tree from inorder and any of the postorder or preorder traversal have been proposed. In this paper one of the proposed algorithms has been examined. This algorithm computes the wrong tree for some input sequences. We show a particular situation in which the algorithm fails and a solution for this situation is proposed. The proposed a modified non-recursive algorithm for reconstructing a binary tree which generates the correct tree otherwise an error has been reported.
A tree is a non-linear data structure for fast storing and retrieval of data in primary memory. It represents data in the form of hierarchical form. Data are stored in a tree i.e. called as a node, in which topmost node is called root and each node has one or more nodes lying on the left or right side of a tree. Except for root node each node has a parent node. The information can be extracted from a tree through various traversal algorithms. Tree traversal means visiting the nodes of a tree at once. In this paper, we are studying different algorithms for tree traversal
BIT, 1989
This paper shows that a binary tree can be constructed from its preorder and inorder traversals in linear time and space.
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