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Chapter 12 binary trees.

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12. 11. Binary Search Trees ¶

12. 11.1. binary search tree definition ¶.

A binary search tree ( BST ) is a binary tree that conforms to the following condition, known as the binary search tree property . All nodes stored in the left subtree of a node whose key value is \(K\) have key values less than or equal to \(K\) . All nodes stored in the right subtree of a node whose key value is \(K\) have key values greater than \(K\) . Figure 12.11.1 shows two BSTs for a collection of values. One consequence of the binary search tree property is that if the BST nodes are printed using an inorder traversal , then the resulting enumeration will be in sorted order from lowest to highest.

Figure 12.11.1: Two Binary Search Trees for a collection of values. Tree (a) results if values are inserted in the order 37, 24, 42, 7, 2, 40, 42, 32, 120. Tree (b) results if the same values are inserted in the order 120, 42, 42, 7, 2, 32, 37, 24, 40.

Here is a class declaration for the BST. Recall that there are various ways to deal with keys and comparing records Three typical approaches are key-value pairs , a special comparison method such as using the Comparator class, and passing in a comparator function . Our BST implementation will require that records implement the Comparable interface.

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12. 11.1.1. BST Search ¶

The first operation that we will look at in detail will find the record that matches a given key. Notice that in the BST class, public member function find calls private member function findhelp . Method find takes the search key as an explicit parameter and its BST as an implicit parameter, and returns the record that matches the key. However, the find operation is most easily implemented as a recursive function whose parameters are the root of a subtree and the search key. Member findhelp has the desired form for this recursive subroutine and is implemented as follows.

Proficient

12. 11.2. BST Insert ¶

Now we look at how to insert a new node into the BST.

Note that, except for the last node in the path, inserthelp will not actually change the child pointer for any of the nodes that are visited. In that sense, many of the assignments seem redundant. However, the cost of these additional assignments is worth paying to keep the insertion process simple. The alternative is to check if a given assignment is necessary, which is probably more expensive than the assignment!

We have to decide what to do when the node that we want to insert has a key value equal to the key of some node already in the tree. If during insert we find a node that duplicates the key value to be inserted, then we have two options. If the application does not allow nodes with equal keys, then this insertion should be treated as an error (or ignored). If duplicate keys are allowed, our convention will be to insert the duplicate in the left subtree.

The shape of a BST depends on the order in which elements are inserted. A new element is added to the BST as a new leaf node, potentially increasing the depth of the tree. Figure 12.11.1 illustrates two BSTs for a collection of values. It is possible for the BST containing \(n\) nodes to be a chain of nodes with height \(n\) . This would happen if, for example, all elements were inserted in sorted order. In general, it is preferable for a BST to be as shallow as possible. This keeps the average cost of a BST operation low.

12. 11.3. BST Remove ¶

Removing a node from a BST is a bit trickier than inserting a node, but it is not complicated if all of the possible cases are considered individually. Before tackling the general node removal process, we will first see how to remove from a given subtree the node with the largest key value. This routine will be used later by the general node removal function.

The return value of the deletemax method is the subtree of the current node with the maximum-valued node in the subtree removed. Similar to the inserthelp method, each node on the path back to the root has its right child pointer reassigned to the subtree resulting from its call to the deletemax method.

A useful companion method is getmax which returns a pointer to the node containing the maximum value in the subtree.

Now we are ready for the removehelp method. Removing a node with given key value \(R\) from the BST requires that we first find \(R\) and then remove it from the tree. So, the first part of the remove operation is a search to find \(R\) . Once \(R\) is found, there are several possibilities. If \(R\) has no children, then \(R\) ’s parent has its pointer set to NULL. If \(R\) has one child, then \(R\) ’s parent has its pointer set to \(R\) ’s child (similar to deletemax ). The problem comes if \(R\) has two children. One simple approach, though expensive, is to set \(R\) ’s parent to point to one of \(R\) ’s subtrees, and then reinsert the remaining subtree’s nodes one at a time. A better alternative is to find a value in one of the subtrees that can replace the value in \(R\) .

Thus, the question becomes: Which value can substitute for the one being removed? It cannot be any arbitrary value, because we must preserve the BST property without making major changes to the structure of the tree. Which value is most like the one being removed? The answer is the least key value greater than the one being removed, or else the greatest key value less than (or equal to) the one being removed. If either of these values replace the one being removed, then the BST property is maintained.

When duplicate node values do not appear in the tree, it makes no difference whether the replacement is the greatest value from the left subtree or the least value from the right subtree. If duplicates are stored in the left subtree, then we must select the replacement from the left subtree. 1 To see why, call the least value in the right subtree \(L\) . If multiple nodes in the right subtree have value \(L\) , selecting \(L\) as the replacement value for the root of the subtree will result in a tree with equal values to the right of the node now containing \(L\) . Selecting the greatest value from the left subtree does not have a similar problem, because it does not violate the Binary Search Tree Property if equal values appear in the left subtree.

Alternatively, if we prefer to store duplicate values in the right subtree, then we must replace a deleted node with the least value from its right subtree.

12. 11.4. BST Analysis ¶

The cost for findhelp and inserthelp is the depth of the node found or inserted. The cost for removehelp is the depth of the node being removed, or in the case when this node has two children, the depth of the node with smallest value in its right subtree. Thus, in the worst case, the cost for any one of these operations is the depth of the deepest node in the tree. This is why it is desirable to keep BSTs balanced , that is, with least possible height. If a binary tree is balanced, then the height for a tree of \(n\) nodes is approximately \(\log n\) . However, if the tree is completely unbalanced, for example in the shape of a linked list, then the height for a tree with \(n\) nodes can be as great as \(n\) . Thus, a balanced BST will in the average case have operations costing \(\Theta(\log n)\) , while a badly unbalanced BST can have operations in the worst case costing \(\Theta(n)\) . Consider the situation where we construct a BST of \(n\) nodes by inserting records one at a time. If we are fortunate to have them arrive in an order that results in a balanced tree (a “random” order is likely to be good enough for this purpose), then each insertion will cost on average \(\Theta(\log n)\) , for a total cost of \(\Theta(n \log n)\) . However, if the records are inserted in order of increasing value, then the resulting tree will be a chain of height \(n\) . The cost of insertion in this case will be \(\sum_{i=1}^{n} i = \Theta(n^2)\) .

Traversing a BST costs \(\Theta(n)\) regardless of the shape of the tree. Each node is visited exactly once, and each child pointer is followed exactly once.

Below is an example traversal, named printhelp . It performs an inorder traversal on the BST to print the node values in ascending order.

While the BST is simple to implement and efficient when the tree is balanced, the possibility of its being unbalanced is a serious liability. There are techniques for organizing a BST to guarantee good performance. Two examples are the AVL tree and the splay tree . There also exist other types of search trees that are guaranteed to remain balanced, such as the 2-3 Tree .

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Programming Assignment #1: CSE 100 (Fall 2014)

Binary search tree.

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Assignment overview, bstiterator.hpp, bstnode.hpp, test_bst.cpp, unit testing, how to find c++ help online, turning in your assignment.

Read Drozdek, Chapter 6 and make sure you understand the basic properties of binary search trees. In this assignment you will implement the basic "insert" and "find" operations, as well as the iterator pattern, using concepts from the C++ Standard Template Library.

Assuming you have successfully completed part 1, you will have your repository that contains the provided code for this assignment. In it you will find the following files:

You will modify the three *.hpp files, providing full definitions of member functions marked with // TODO , and turn them in. In completing the implementation, you shouldn't remove anything from those files (except the // TODO comments!). If design considerations dictate it, you can add additional functions or variables, but they should be in a private: section, indicating that those additions are part of the implementation, and not part of the public or protected interface to these classes.

The test_BST.cpp file is a simple C++ application which partially tests some aspects of the BST class. You should add code here, in order to test your program as thoroughly as you can (We will test it as thoroughly as we can when grading it!).

The Makefile is provided for your convenience. Running make in the directory will compile the test_BST.cpp application, creating an executable named bst which can be run from the command line. Modify this Makefile as needed, or create your own, to support your development process. Of course, it is possible that the provided Makefile works perfectly fine for your purposes, and won’t need to be changed.

Your first task is to understand the code. Do not attempt to start writing code until you have a firm grasp of what the provided code does and what you are supposed to do with it. Nearly all the required functionality is specified in the code and the comments, so you'll need to read everything carefully. Unlike in previous classes (such as certain offerings of CSE8A and 8B) we are not providing a lot of scaffolding to walk you through what this code does, outside of what we have already provided in class. This is a gateway course, and in the upper division you will at times be given vague assignment specifications and expected to make sense of them. We've tried to provide some guidance, but not as much as you may have received in previous classes. We also encourage test-driven development here. Make sure you can understand the provided test file, and then add your own tests before you start implementing any code of your own.

If you have any questions about the assignment, please come to the lab during tutor hours, visit a TA during their office hours, or post on the discussion forum. Tutor and TA hours are posted on the main course website.

If you do not understand what you are doing, you will not learn anything from this assignment and the whole point of it will be lost. We are happy to help, but you must seek this help. And START EARLY. Read the code RIGHT NOW!

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October 17 @ 11:00 PM

This means your solution to the assignment must be turned in by the due deadline. The turn-in procedure is described below in the section Turning in your assignment . It is highly recommended that you start this assignment immediately; terminals and tutors can get very busy at the last minute. The files that contain your solution to the assignment must be turned in by the due deadline. NO EXCEPTIONS! You may develop your code anywhere, but you must ensure it compiles and runs correctly on ieng6, because that’s where we’ll be testing it.

In doing this assignment, it's your responsibility to understand the course rules for integrity of scholarship.

Acquiring the Starter Files

You must follow these steps to acquire the starter code and prepare for the assignment. Do not directly copy files from any directory that you find on ieng6, as may have been the procedure in previous offerings of this course!

Instead, see Using Gitlab for the proper steps.

The bst.hpp file will be used to maintain the binary search tree structure and functionality.

In the bst.hpp file, you will notice the line template<typename Data> preceding the class BST { (the start of the class definition). This may be something you have never seen before and is referred to as a template class. Templates in C++ are a powerful feature allowing you to define a pattern which the compiler can turn into actual code, essentially allowing the class to work with generic types. REMEMBER: A template is a pattern, not the actual implementation. At compile time, the compiler will translate the pattern into actual code for each unique type used with the pattern. A BST that holds doubles and a BST that holds integers will result in two separate BST classes, one for each type. Without features like templates, one would have to create a new class for each data type to be stored in the BST manually.

The word Data following typename is the type identifier . When reviewing the source code, you will see Data used everywhere the object type to be stored in or retrieved from the BST is referred to.

Note: Since the BST class is a template class, instantiation requires some special notation. The compiler needs to know the type to be used with the object being created. Creating an instance of the BST template class is done as follows: class_name<data_type> identifier . For example, to create an instance of a BST class that will hold integers, you would type the following: BST<int> myIntBst;

When reviewing the code, you will notice the following in several places: std::pair<iterator, bool> . The :: characters separating std and pair is known as a scope resolution operator. In C++, the functionality that is part of the standard template library is defined inside the std namespace. Doing some searching on C++ scope resolution operator will provide plenty of information on its usage.

Returning to std::pair<iterator, bool> , this is an example of a C++ template class, except that it supports two unique types, unlike the BST which only supports one. For template classes having support for multiple types, separate them with a comma. The pair , defined in the std namespace, is defined inside the utility header (part of the standard template library). For more information on the pair class, see: C++ std::pair .

What you need to know for this assignment is that a pair object is created via std::make_pair and its members are referred to as first and second. For example, if you created a pair named myPair, then to refer to the first and/or second values, you would type myPair.first and myPair.second, respectively. Note, if myPair was a pointer, you would use myPair->first and myPair->second instead.

Another part of the source code needing some explanation is virtual ~BST() . C++ is not a managed programming language like Java, Python, C#, etc. In other words, it does not have a garbage collector responsible for cleaning up resources. This means YOU are the garbage collector. To facilitate this, C++ introduces the destructor . Note: the virtual keyword is not part of the destructor. For this assignment, virtual serves no purpose. When an object created on the stack goes out of scope, the destructor is automatically invoked. However, when an object is created on the heap with the new keyword, the memory must be freed with the delete keyword. Not only will delete free the memory used, but it will also invoke the destructor . In this area of the code, you will be responsible for deleting each of the nodes created and stored in the tree. Nodes will be explained in the bstnode.hpp section.

The bstiterator.hpp file shall be responsible for providing forward iteration from any point in the BST.

Note: Concepts already discussed in the other sections will not be mentioned again. If anything was not explained, it was probably covered in a previous section.

The first part of the source code to note: class BSTIterator : public std::iterator<std::input_iterator_tag,Data> { . C++ supports private, protected, and public inheritance. For understanding the differences, see: C++ class inheritance . The main area of interest is the access permissions of subclasses. Essentially, this line is sub-classing from the std::iterator class defined in the C++ standard template library. See: std::iterator .

Sub-classing from std::iterator ensures we do this in a way common with built-in C++ containers. Also see: Input Iterator .

When looking at the functions defined in this file, you will see operator++() and operator++(int) , which overload the pre-increment and post-increment operators respectively. C++ supports operator overloading, allowing the programmer to define the behavior of operators such as pre- and post-increment. Note the int parameter in the second version above. The function does not require an int as an argument, this is the convention used to differentiate between pre- and post-increment. The support for operator overloading essentially allows the programmer to type ++myObject instead of myObject.increment() .

Another feature of C++ exposed in this file is the friend keyword. Any class or function defined as a friend of a class has access to ALL the non-public members of the class. Since classes BST and BSTNode are defined as friends of class BSTIterator, they have access to the private constructor and instance variables. This sort of design allows the friend classes to create iterator objects, but no one else.

The bstnode.hpp file shall be responsible for holding the value to be stored in the tree along with a link to its parent and children nodes.

From lecture and the reading, it should be clear that a BST is a special type of tree that maintains a well-defined ordering property, allowing for fast searches. It is this ordering property that enables efficient support for the successor method -- which you will be implementing. A sound understanding of this ordering property is necessary to correctly implement the successor method.

Much of this class should be self-explanatory by now, however, one part needs a mention, which will be useful in testing your program. Near the end of the file you will see the following template function: std::ostream & operator . This is a helper function designed to allow the details of a node to be printed using the stream extraction operator << . It's usage is as follows. Assume a BSTNode<int> object exists whose identifier is myNode . To print the properties of this node to standard output, you would do the following: std::cout << myNode .

The test_bst.cpp file provides a starting test suite for validating your binary search tree's correctness. It is by no means exhaustive and you should add additional functionality to this file to further test your program. Review the contents to see what tests are being performed and how they are implemented. This should help you identify what has not been tested and what functionality needs additional testing.

When writing your assignment, generally, it's best to first implement the general cases. For example, when implementing the insert operation for your BST, start with the assumption that a root node already exists and begin the insert logic. You can even have your constructor initially create a root node. Once you have this logic working, then consider the special logic that might be needed for the initial insertion of a node. Think about the possible situations under which insert can be called. For example, a new tree with zero elements, a tree with only one element, a tree with a root and left child only, and a tree with a root and right child only. Anything else is just another case of the above.

Probably the most tricky parts of this assignment will be implementing the destructor in the BST class and the successor function in the BSTNode class. As mentioned earlier, C++ does not manage resources for you. This mean you will have to delete each node that you created and attached to the tree in the destructor. It is suggested that you work this logic out on paper first. Think about how you can traverse a tree deleting nodes as you go. Once you solve this problem, implementing the destructor becomes easy. Work this out on paper with example trees. Think about the different tree shapes you can encounter and make sure your logic works regardless.

Equally complex is the successor method. Again, you should draw several trees on paper, thinking about where the successor node would exist given a certain position in the tree. As an example, draw a tree inserting the following nodes in this order { 100, 50, 75 }. Given you are currently looking at 75, who is the successor? How would you navigate to that node (100) in this case. Create more trees with differing shapes considering all the possible logic that is needed given the position of a node in the tree. Remember, you don't need to get too complicated. In most cases, a tree with 5-7 nodes will present all the different shapes. Trees with more nodes are really just more instances of the same problem.

Tip: Keep your code as simple as possible. Many a times the recursive algorithm is much easier to implement than the iterative one. Also, it's usually easier to read and debug.

While many of you may be inclined to perform a quick online search for a specific compiler error or the usage of a C++ feature, this will not be helpful in the long run. It might be faster to search stack overflow for an answer, but you should really learn how to utilize and read technical documentation. You will not truly learn how certain parts of the C++ standard template library are intended to be used any other way. In addition, reading this documentation will help YOU write better technical documentation and comments in your code.

For starters, you should always have a copy of the C++ operator precedence table available when you are programming. Not only does this help you understand how expressions are evaluated, but it will prevent you from cluttering your code with unnecessary parenthesis. See: C++ Operator Precedence . Also, a great resource for what is available in the standard template library and how to properly use these resources is available here: cppreferene.com .

Before searching online for your problem, take a moment to look at the error, review your code and logic, and be a visual debugger. You will learn more about compiler errors and what they mean by doing this. Another very useful tip is to intentionally do something in code incorrectly and then try to compile it. Study the compiler errors for something you know is wrong. This way you know how the compiler is informing you of the problem.

When you have completed the assignment, you must submit it. You will do this by pushing your changes to GitLab. Follow these steps, also go through the git instructions link provided in the Part 1 of this lab (above):

  • Run a ’git status’ to see what files have been changed.
  • Use ’git add <files>’ to stage certain files for a commit, use ’git add .’ to add all files in the working directory to the commit (hopefully you will not be doing this at this late stage—all files should already be in your repository!)
  • Use ’git commit -m "FINAL SUBMISSION"’ to commit the changes to your repository and indicate that you are done and this is your final submission.
  • To push your changes onto your repository on GitLab run ’git push origin master’ Don’t forget this push! You must do this to get credit.
  • As an added safety check, you should log on to GitLab on the web and verify that your push went through.

Some points to keep in mind:

  • If you are submitting with a partner, just push to the joint repository on behalf of both partners. That’s your joint submission.
  • Use the commit message "FINAL SUBMISSION" in ONLY ONE of your individual and partner repositories. You shouldn’t be using both anyway, but if you happen to have pushed changes to both, just make sure only one contains the "FINAL SUBMISSION" commit message.
  • If you decide you want to resubmit your assignment before the deadline, just do another commit with the message "FINAL SUBMISSION" (and don’t forget to push to GitLab!!). We will grade the last commit before the deadline (or the slip day deadline if you use slip day(s)) that is tagged as the "FINAL SUBMISSION".
  • If you submit the assignment within 24 hours after the deadline (i.e. if there is a commit tagged "FINAL SUBMISSION") you will be charged a slip day. If it is more than 24 but within 48 hours, you will be charged 2 slip days. If you are out of slip days, or after 48 hours, we’ll roll back to the last commit tagged as final that was submitted before the deadline.

There are 25 possible points on the assignment. If your solution files do not compile and link error-free, you will receive 0 points for this assignment. We will compile and test your program using the g++ compiler and runtime environment on ieng6. To get full credit, style and comments count.

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Introduction to Binary Tree – Data Structure and Algorithm Tutorials

A binary tree is a tree data structure in which each node can have at most two children, which are referred to as the left child and the right child. 

The topmost node in a binary tree is called the root, and the bottom-most nodes are called leaves. A binary tree can be visualized as a hierarchical structure with the root at the top and the leaves at the bottom.

Binary trees have many applications in computer science, including data storage and retrieval, expression evaluation, network routing, and game AI. They can also be used to implement various algorithms such as searching, sorting, and graph algorithms.

Representation of Binary Tree:

Each node in the tree contains the following:

  • Pointer to the left child
  • Pointer to the right child

Binary Tree

In C, we can represent a tree node using structures. In other languages, we can use classes as part of their OOP feature. Below is an example of a tree node with integer data.

Basic Operations On Binary Tree:

  • Inserting an element.
  • Removing an element.
  • Searching for an element.
  • Deletion for an element.
  • Traversing an element. There are four (mainly three) types of traversals in a binary tree which will be discussed ahead.

Auxiliary Operations On Binary Tree:

  • Finding the height of the tree
  • Find the level of the tree
  • Finding the size of the entire tree.

Applications of Binary Tree:

  • In compilers, Expression Trees are used which is an application of binary trees.
  • Huffman coding trees are used in data compression algorithms.
  • Priority Queue is another application of binary tree that is used for searching maximum or minimum in O(1) time complexity.
  • Represent hierarchical data.
  • Used in editing software like Microsoft Excel and spreadsheets.
  • Useful for indexing segmented at the database is useful in storing cache in the system,
  • Syntax trees are used for most famous compilers for programming like GCC, and AOCL to perform arithmetic operations.
  • For implementing priority queues.
  • Used to find elements in less time (binary search tree)
  • Used to enable fast memory allocation in computers. 
  • Used to perform encoding and decoding operations.
  • Binary trees can be used to organize and retrieve information from large datasets, such as in inverted index and k-d trees.
  • Binary trees can be used to represent the decision-making process of computer-controlled characters in games, such as in decision trees.
  •  Binary trees can be used to implement searching algorithms, such as in binary search trees which can be used to quickly find an element in a sorted list.
  • Binary trees can be used to implement sorting algorithms, such as in heap sort which uses a binary heap to sort elements efficiently.

Binary Tree Traversals:

Tree Traversal algorithms can be classified broadly into two categories:

  • Depth-First Search (DFS) Algorithms
  • Breadth-First Search (BFS) Algorithms

Tree Traversal using Depth-First Search (DFS) algorithm can be further classified into three categories:

  • Preorder Traversal (current-left-right): Visit the current node before visiting any nodes inside the left or right subtrees. Here, the traversal is root – left child – right child. It means that the root node is traversed first then its left child and finally the right child.
  • Inorder Traversal (left-current-right): Visit the current node after visiting all nodes inside the left subtree but before visiting any node within the right subtree. Here, the traversal is left child – root – right child.  It means that the left child is traversed first then its root node and finally the right child.
  • Postorder Traversal (left-right-current): Visit the current node after visiting all the nodes of the left and right subtrees.  Here, the traversal is left child – right child – root.  It means that the left child has traversed first then the right child and finally its root node.

Tree Traversal using Breadth-First Search (BFS) algorithm can be further classified into one category:

  • Level Order Traversal:  Visit nodes level-by-level and left-to-right fashion at the same level. Here, the traversal is level-wise. It means that the most left child has traversed first and then the other children of the same level from left to right have traversed. 

Let us traverse the following tree with all four traversal methods:

Binary Tree

Pre-order Traversal of the above tree: 1-2-4-5-3-6-7 In-order Traversal of the above tree: 4-2-5-1-6-3-7 Post-order Traversal of the above tree: 4-5-2-6-7-3-1 Level-order Traversal of the above tree: 1-2-3-4-5-6-7

Implementation of Binary Tree:

Let us create a simple tree with 4 nodes. The created tree would be as follows. 

Binary Tree

Below is the Implementation of the binary tree:

Conclusion:

Tree is a hierarchical data structure. Main uses of trees include maintaining hierarchical data, providing moderate access and insert/delete operations. Binary trees are special cases of tree where every node has at most two children.

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Assignment 11

This assignment is an exercise in implementing the map ADT using a binary search tree. You will need to write a template structure to represent a tree node, and a template class to represent the tree.

1. Initial Setup

Log in to Unix.

Run the setup script for Assignment 11 by typing:

2. Files You Must Write

You will need to write one template structure and one template class for this assignment. A main program to test your class will be provided.

Since this is a C++ template, all of the code for both the structure and the class should be placed in a single file, bstree.h . That includes both structure / class definitions as well as the definitions for all of the functions. See the Hints section for an outline of how to structure this file and the order that things need to be coded.

The node structure

Data members

This template structure should have four data members: a member of the template parameter type used to store a key, a member of the template parameter type T used to store a value, a pointer to a node<K, V> that points to the node's left subtree, and a pointer to a node<K, V> that point's to the node's right subtree.

Member functions

Constructor

The structure should have a constructor that can be used to initialize the data members of the tree node.

The bstree class

This class should have two data members. The first should be a pointer to a node<K, V> that will point to the root node of the tree (or be nullptr if the tree is empty). The second should be a size_t variable that will be used to store the tree size, the number of elements or values currently stored in the binary search tree.

The bstree class should implement the following member functions.

template <class K, class V> bstree<K, V>::bstree()

The class should have a default constructor that sets the root pointer data member of the tree to nullptr and the tree size to 0.

template <class K, class V> bstree<K, V>::~bstree()

The class should have a destructor. The destructor can simply call the clear() method.

template <class K, class V> bstree<K, V>::bstree(const bstree<K, V>& x)

The class should have a proper copy constructor. Making a copy of the tree nodes can be done by performing a modified preorder traversal as described in the course notes on binary search trees.

template <class K, class V> bstree<K, V>& bstree<K, V>::operator=(const bstree<K, V>& x)

The assignment operator should be properly overloaded.

template <class K, class V> void bstree<K, V>::clear()

This member function should set the tree back to the empty state, deleting all of the nodes in the tree and setting the size back to 0.

template <class K, class V> size_t bstree<K, V>::size() const

Returns the tree size.

template <class K, class V> size_t bstree<K, V>::height() const

Returns the tree height. In this assignment, we will use a slightly modified definition for the height of a tree. An empty tree will have height 0, a tree with only a root node will have height 1. The height of any other node in the tree is the height of its parent node + 1. The height of the tree can then be defined as the maximum height of any of its nodes.

template <class K, class V> bool bstree<K, V>::empty() const

Returns true if the tree size is 0; otherwise, returns false.

template <class K, class V> const K& bstree<K, V>::min() const

This member function should return the minimum key in the tree. You may assume that this function will not be called for an empty tree.

template <class K, class V> const K& bstree<K, V>::max() const

This member function should return the maximum key in the tree. You may assume that this function will not be called for an empty tree.

template <class K, class V> bool bstree<K, V>::insert(const K& key, const V& value)

This member function should attempt to insert a key and value into the binary search tree. If the key already exists in the tree, the function should return false. Otherwise, a new tree node containing the key and value should be inserted in the correct spot to maintain the ordered property of the binary search tree, the size of the tree should be incremented, and the function should return true.

template <class K, class V> bool bstree<K, V>::remove(const K& key)

This member function should attempt to remove the specified key from the binary search tree. If the key is not in the tree, the function should return false. Otherwise, the node with a matching key should be removed, the size of the tree should be decremented, and the function should return true.

template <class K, class V> const node<K, V>* bstree<K, V>::find(const K& key) const

This member function should attempt to find the specified key in the binary search tree. If the key is not in the tree, the function should return nullptr . Otherwise, it should return the address of the node that contains the specified key.

template <class K, class V> void bstree<K, V>::preorder() const

This member function should perform a preorder traversal of the tree from left to right. As each node is visited, it should have its key and value printed (see Output below for the required format).

template <class K, class V> void bstree<K, V>::inorder() const

This member function should perform a inorder traversal of the tree from left to right. As each node is visited, it should have its key and value printed (see Output below for the required format).

template <class K, class V> void bstree<K, V>::postorder() const

template <class K, class V> void bstree<K, V>::level_order() const

This member function should perform a level order traversal of the tree from left to right. As each node is visited, it should have its key and value printed (see Output below for the required format).

3. Files We Give You

The setup script will create the directory Assign11 under your csci241 directory. It will copy a makefile named makefile to the assignment directory.

You will also receive a driver program named assign11.cpp which contains a main() function that will test your bstree class.

The correct output for this assignment is shown below:

If you would like a copy of this output to compare against your own program's output using the diff command, it is available on Unix at the pathname /home/turing/t90kjm1/CS241/Output/Spring2022/Assign11/output11.txt .

The driver program is designed to make this assignment easy to develop incrementally. Simply comment out all of the lines of the driver program that call functions that you haven't written yet. You should be able to write, test, and debug function at a time.

This assignment sheet describes the "public interface" for the binary search tree class. You are welcome (and encouraged) to write additional private member functions for your class as needed.

Pseudocode for many of the tree member functions can be found in the notes on Binary Tree Traversals and Binary Search Trees on the course web site.

CSE333—Systems Programming

Programming Assignment #2

due : Thursday, 10/20/11, 11:59 pm

In this programming assignment you will practice writing a complete C++ class that includes operator overloading.   You are building a class for storing a set of simple int values.   There is a built-in set class available in C++, but we are going to write our own from scratch using a binary search tree.   As a result, you will get practice with dynamic memory management and will learn how to work with both pointers and references.

As noted above, you will store each set of integers using a binary search tree so that we can get fast access and add times.   We aren’t going to worry about balancing the tree, so there will be degenerate cases where the structure is not efficient, but we’ll live with that.   But when we construct trees ourselves, we will be careful to keep them balanced when we can.

A skeleton version of the class is provided on the class web page that includes the definition of a nested class called TreeNode that you should use in constructing your binary search tree.   Your class will keep track of the tree and should also store the size so that it can be returned quickly rather than computed.   Remember that one difference between Java and C++ is that you can’t use the same identifier for both an instance variable and a member function.   So if there is a member function called size, you can’t also have an instance variable called size.

Your class should have the following constructors:

·          A constructor that takes an int as a parameter.   The parameter indicates an initial size for the set.   If the client asks for a set of size n, then you should construct a set containing the numbers 0 through (n-1).   You should construct this as a balanced tree with extra nodes going in the left subtree when a subtree contains an even number of nodes.   The construction of this tree should take no more than O( n) time.   The default value for the parameter should be 0 so that by default it constructs an empty set.

·          A constructor that takes a sorted vector<int> as a first parameter.   As with the first constructor, you should construct a balanced tree from this vector, putting extra values in the left subtree when a subtree has an even number of elements, and running in O( n) time.   You do not have to handle errors if the vector is not sorted or if it contains duplicates.

·          A copy constructor that makes a copy of an existing set.   It should run in O( n) time where n is the size of the set.

You should make sure that it is not legal to assign a set to a simple int or to a vector without calling a constructor directly.

Your class should have the following accessors:

·          a size method that returns the current number of elements in the set

·          a contains method that takes an int as a parameter and that returns whether or not the set contains that value (this should run in O(log n) time assuming the tree is balanced)

·          a print method that takes an ostream as a parameter and that prints to the stream a bracketed, comma-separated list of values in sorted order, as in [3, 18, 42].   There should be a space after each comma, but otherwise there should be no extraneous spaces.

·          a toVector method that returns a vector<int> containing the values from the set in sorted order

Your class should have the following mutators:

·          an add method that takes an int value to add to the set.   This should perform the classic binary search tree add to include this value in the tree.   If the value is already in the tree, then the set should be unchanged so that we avoid having duplicates.

·          a clear method that removes all values from the set

Both of these methods should return a reference to the implicit parameter so that calls on the mutator can be chained together, as in:

s.add( 8).add(9).add(17).add(14).add(92);

s.clear( ).add(3).add(14);

In addition, your IntSet should overload the following operators:

·          the insertion operator (<<) that allows us to write code like “cout << s”

·          the assignment operator that allows us to write code like “s1 = s2”

·          the addition operator for a set/int combination in either order that returns a new set that contains the old set values with the new value added to it

·          the addition operator for two sets that returns a new set that is the union of the two sets

·          the multiplication operator for two sets that returns a new set that is the intersection of the two sets

·          a += operator that allows us to add either an int or a set to a set

You are allowed to introduce additional member functions, but they should be declared private.   You are likely to introduce many such functions for manipulating the tree.

Computing the union and intersection can lead to a highly unbalanced tree.   You should solve this problem by converting the two sets involved into vectors and then using those vectors to construct a third vector that contains the result.   You can then use the constructor that takes a vector to construct the result.   These operations are required to run in O( n + m) time where n and m are the sizes of the two sets you are working with.   The += operator for adding a set to a set can also lead to an unbalanced tree, but it can be defined in terms of the union operator to avoid this problem.

This trick of converting the tree to a vector is tempting, but don’t use it for other member functions because it is a bit of a hack.   For example, you shouldn’t use this trick to implement the print function or the copy constructor.

As in the TreeNode class that is included in the skeleton file, you will refer to nodes using a pointer to a TreeNode of type “TreeNode *”.   Passing such values as parameters is a bit tricky in C++ and there are some associated subtleties that you don’t need to entirely understand.   For our purposes, you should distinguish two kinds of parameters.   If you want an alias for the pointer so that you can change its value, then declare the parameter in this manner:

TreeNode * & root

If you want to pass a pointer in a read-only way so that the tree can be examined but not modified, then declare the parameter in this manner:

const TreeNode * const & root

If you are working with one of these read-only variables, then local variables would have to be declared with a similar constraint, as in:

const TreeNode * temp = root;

Oddly enough, such a variable can be changed.   For example, you could use it in a loop that resets temp to temp->left or temp->right.   The const here means that it can’t be used to change the contents of the tree (you wouldn’t, for example, be able to change temp->data).

You are expected to use appropriate parameter types and return types for all of your member functions and the overloaded operators.   Use the Weiss Rational class as a guide.   You are also expected to use good programming style and to comment all functions appropriately, including relevant preconditions.

You should use valgrind to verify that your class does not produce a memory leak.   A substantial number of points will be awarded for proper memory management.

You should split your class into two files: IntSet.h and IntSet.cpp and turn in both files.   All member functions should be defined in the cpp file.   The only function that should have a definition in the header file is the constructor for the TreeNode class.   Your header file should be written in the ifndef/define/endif form so that it would not be loaded more than once by client programs.

  • Part 1: Binary Tree Class
  • Part 2: Binary Search Tree Class
  • Part 3: AVL Tree

Constructors

  • BinaryTree() – constructs an empty binary tree.
  • BinaryTree( T *elements, int n_elements ) – constructs a complete tree having elements , in the usual order of a complete binary tree. Note: the code implementing this is provided.
  • BinaryTree( const BinaryTree& src ) – copy constructor; constructs a duplicate of src
  • ~BinaryTree() – destructor (deletes all the nodes in this tree).
  • bool is_empty() – returns true if this tree is empty, and false otherwise
  • int node_count() const – returns the total number of nodes in this tree
  • int leaf_count() const – returns the number of leaf in this tree
  • int height() const – returns the height of this tree: the height of the empty tree is 0; the height of a nonempty tree is one plus the longest path from the root to any leaf.
  • bool empty_this() – empties (and deallocates) this tree
  • int to_flat_array( T* nodes, int max ) const – copies the node elements to the nodes array assuming this is a complete binary tree. Note: the code implementing this is provided.
  • void preorder( void (*f)(const T&) ) const – Performs a preorder traversal of this tree, applying f on the element of each node visited.
  • void inorder( void (*f)(const T&) ) const – Performs an inorder traversal of this tree, applying f on the element of each node visited.
  • void postorder( void (*f)(const T&) ) const – Performs a postorder traversal of this tree, applying f on the element of each node visited.
  • bool operator==( const BinaryTree& src ) const – returns true if this and src are identical; that is, if the tree structures are the same and the corresponding elements match.
  • bool operator!=( const BinaryTree& src ) const – logical complement of the == operator.
  • BinaryTree& operator=( const BinaryTree& src ) const – assignment operator: copies src to this tree (removing the prior content).

Input/Output

  • template<class S> friend ostream& operator< & src ); – writes the sequence of nodes of src in complete-tree order to out assuming this is a complete binary tree. Note: the code implementing this has been provided.
  • BinarySearchTree() – constructs an empty binary search tree.
  • BinarySearchTree( T *elements, int n_elements ) – constructs a binary search tree by sequentially inserting elements from the elements array (assuming elements contains n_elements ). Note that this constructor is fundamentally different from that in the BinaryTree class, which treated the input as elements in a complete binary tree.
  • BinarySearchTree( const BinaryTree& src ) – copy constructor; constructs a duplicate of src . (This operation is identical to the construction of a general binary tree, so the BinaryTree copy constructor is called directly in the listing above.)
  • ~BinaryTree() – destructor
  • bool insert( T elem ) – inserts elem into this tree, returning true on success, and false otherwise (i.e., if an element having the same key as elem exists in the tree).
  • lookup( K key ) – searches this tree for an element having key key , and returns a pointer to the element (not the node) in which it is found, or NULL if it is not found. The type parameter K is the type of the key, which need not be the same as the type T of the node elements. However, you can assume that = and < operators exist for comparing values of type K with values of type T (see the Remarks page below).
  • init( T* elements, int n_elements ) – initializes this tree by inserting each each element from the elements array in sequence. As this is an initialization, the tree should be emptied first.
  • init_from_sorted_array( T* elements, int n_elements ) – initializes this tree from the elements array, assuming the array is sorted. The resulting tree must be balanced. In order to do this, you will have to insert the elements non-sequentially, as discussed in class. Again, the first thing this function should do is empty the tree. (Properly implementing this function is a major part of this assignment.)
  • to_array( T* elements, int max ) const – copies the elements from this tree to the elements by way of an inorder traversal. At most max elements are copied, and the return value is the total number of elements.
  • operator>> reads a string of elements, inserting them into this tree in the order read. Note: this method does not empty the tree first, so that multiple >> operations continue to grow the tree.
  • operator<< writes the nodes of this tree in sorted order (i.e., via an inorder traversal).

COMMENTS

  1. c++

    Assignment Operator: template<typename Type> BST<Type>& BST<Type>::operator= (const BST& that) { if (this != &that) { this->clear (); Node *c = that.root; preORet (c); } return *this; } Recursive Function Called:

  2. A binary tree , its copy constructor and assignment operator

    Assignment operator. The assignment operator is not 100% exception safe. You should not modify the current object while you have not yet finished making the copy. Node& Node::operator= (const Node& other) { value = other.value; Node * left_orig = left; left = new Node (*other.left); delete left_orig; // You have modified the left side.

  3. Binary Tree Data Structure

    Introduction Basic Operation Traversals Standard Problems on Binary Trees Introduction : Introduction to Binary Tree - Data Structure and Algorithm Tutorials Properties of Binary Tree Types of Binary Tree Applications, Advantages and Disadvantages of Binary Tree Binary Tree (Array implementation) Complete Binary Tree Perfect Binary Tree

  4. PDF Binary Search Trees

    A binary search tree (or BST) is a data structure often used to implement maps and sets. The tree consists of a number of nodes, each of which stores a value and has zero, one, or two children.

  5. 12. 11. Binary Search Trees

    12. 11.1. Binary Search Tree Definition¶. A binary search tree (BST) is a binary tree that conforms to the following condition, known as the binary search tree property.All nodes stored in the left subtree of a node whose key value is \(K\) have key values less than or equal to \(K\).All nodes stored in the right subtree of a node whose key value is \(K\) have key values greater than \(K\).

  6. PDF Lecture 6: Binary Trees I

    Tree Navigation. Find first node in the traversal order of node <X>'s subtree (last is symmetric) If <X> has left child, recursively return the first node in the left subtree. Otherwise, <X> is the first node, so return it. Running time is O(h) where h is the height of the tree. Example: first node in <A>'s subtree is <F>.

  7. C++ Assignment Operator Overloading

    Practice Prerequisite: Operator Overloading The assignment operator,"=", is the operator used for Assignment. It copies the right value into the left value. Assignment Operators are predefined to operate only on built-in Data types. Assignment operator overloading is binary operator overloading.

  8. Basic Operations on Binary Tree with Implementations

    A binary tree is a tree that has at most two children. The node which is on the left of the Binary Tree is called "Left-Child" and the node which is the right is called "Right-Child". Also, the smaller tree or the subtree in the left of the root node is called the "Left sub-tree" and that is on the right is called "Right sub-tree".

  9. Binary search tree implementation with Rule of Three

    c++11 Consider adding a move constructor and a move assignment operator to your class. The cost of moving a tree is trivial compared to what it costs to copy a whole tree: template<class T> Tree<T>::Tree (Tree&& other): root (other.root) { other.root = nullptr; } See, all the move constructor does is acquire the memory managed by other and ...

  10. PDF CS271: Data Structures Binary Search Trees

    Binary Search Trees 1. Implement a binary search tree class. Name the class BinarySearchTree and make it templated. You will eventually implement the following methods: copy constructor default constructor destructor assignment operator insert inOrderWalk search max min predecessor successor remove 2. You will need a node class to hold items in ...

  11. Binary Tree

    5. Skewed Binary Tree A skewed binary tree is a pathological/degenerate tree in which the tree is either dominated by the left nodes or the right nodes. Thus, there are two types of skewed binary tree: left-skewed binary tree and right-skewed binary tree. Skewed Binary Tree 6. Balanced Binary Tree

  12. Programming Assignment #1: CSE 100 (Fall 2014)

    Read Drozdek, Chapter 6 and make sure you understand the basic properties of binary search trees. In this assignment you will implement the basic "insert" and "find" operations, as well as the iterator pattern, using concepts from the C++ Standard Template Library. ... C++ supports operator overloading, allowing the programmer to define the ...

  13. Introduction to Binary Tree

    Basic Operations On Binary Tree: Inserting an element. Removing an element. Searching for an element. Deletion for an element. Traversing an element. There are four (mainly three) types of traversals in a binary tree which will be discussed ahead. Auxiliary Operations On Binary Tree: Finding the height of the tree Find the level of the tree

  14. CS 311 Fall 2009: Assignment 7

    Copy assignment operator. Copies a tree, as usual. The copy must contain completely separate data. Dctor. Does all necessary deallocation, destruction, and other necessary clean-up. ... You do not need to implement the Binary Search Tree delete operation. I will be discussing ideas for writing this assignment in class on Friday, November 20.

  15. Assignment 11

    The class should have a proper copy constructor. Making a copy of the tree nodes can be done by performing a modified preorder traversal as described in the course notes on binary search trees. template <class K, class V> bstree<K, V>& bstree<K, V>::operator=(const bstree<K, V>& x) The assignment operator should be properly overloaded.

  16. CSE333 handout #3

    In this programming assignment you will practice writing a complete C++ class that includes operator overloading. You are building a class for storing a set of simple int values. There is a built-in set class available in C++, but we are going to write our own from scratch using a binary search tree.

  17. CSS 343-A Binary Tree Programming Assignment

    The assignment is divided into three parts: Part 1: Binary Tree Class Part 2: Binary Search Tree Class Part 3: AVL Tree Part 1: Binary Tree Class Implementation For this first part, you are asked to complete a C++ implementation of a general binary tree class. This is not specifically a binary search tree (that is in Part 2).

  18. c++

    79 6 _left = other._left; performs a simple pointer assignment since the type of _left is BSNode*. In order to perform an actual reassignment, you'll need to invoke operator= on the nodes themselves - nanofarad Jan 13, 2020 at 0:38

  19. Binary Tree Traversal in Ruby and JavaScript

    Binary Trees are a fundamental data structure in computer science, used for searching, sorting, and as a building block for more complex structures. Traversing a binary tree is a core operation, allowing us to visit each node in the tree exactly once. The order in which we visit these nodes defines the type of traversal.

  20. Trying to overload a Binary Search Tree assignment operator

    1 BinaryTree *newTree; newTree->insert (0); What did you expect this would do? newTree is an indeterminate pointer that you dereference immediately after declaration. You shouldn't be using a pointer there at all, and I suggest you read up on The Rule of Three/Five/Zero, as well as the copy/swap idiom - WhozCraig Nov 1, 2018 at 2:35