CS271: Data Structures. Binary Search Trees


 Reynold Herbert Singleton
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1 CS271: Data Structures Binary Search Trees This project has three parts: an implementation of a binary search tree class, a testing plan, and an experiment in tree height. 1. PART 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 the search tree. Use the following node class exactly. template<class T> struct Node { Node * left; Node * right; Node * parent; T item; }; 3. Use the following public interface for your BinarySearchTree class. You will have to determine the private methods and data that you need to implement the class. template<class T> class BinarySearchTree { public: BinarySearchTree ( void ); BinarySearchTree ( BinarySearchTree & bst ); BinarySearchTree operator= ( BinarySearchTree & bst ); BinarySearchTree ( void ); void insert ( const T item ); 1
2 void inorderwalk ( void ); Node<T> *search ( const T item ); Node<T> *max ( void ); Node<T> *min ( void ); Node<T> *predecessor ( Node<T> *ptr ); Node<T> *successor ( Node<T> *ptr ); void remove ( Node<T> *ptr ); int height ( void ); private: }; 4. The inorderwalk method prints out the tree all on one line, separating tree entries by a single space. 5. The min and max methods return a pointer to the minimum and maximum items (actually a pointer to the nodes that hold those items) in the tree respectively. They should return NULL only if the tree is empty. 6. The search method takes an item of the templated type as input and searches the tree for a copy of that item. It returns a pointer to a node for that item if found or returns NULL if that item is not in the tree. Note it is possible to insert multiple copies of the same item in the tree. Search should return a pointer to one of the items, but it is not defined as to which it has to be. 7. The insert method inserts an item in the tree. The remove method removes the item pointed to by the node pointer argument. The remove method is, by far, the most difficult method of all as it has a number of tricky cases and involves intricate pointer manipulations. 8. The successor and predecessor methods take a node pointer as input and return the node pointer to the item that is the successor or predecessor to the input item. They should return NULL only if the input item is the max or min respectively. Each may return a pointer to a node of the same value if there are multiple copies of that value in the tree; either way, successive calls to each method should eventually move beyond the copies of that shared value. 9. The height method returns an integer indicating the height of the tree (see below for a definition). 10. I recommend to implement the copy constructor and assignment operator near the end. They have some complexity which may be easier to solve if you have some of the other 2
3 methods done first. Note: a recursive solution can be very compact and efficient (avoids a big messy block of code). It may be helpful to define some methods recursively rather than iteratively. Thus you might make a decision to create a private/internal method which is essentially a recursive copy of one of the methods. You should keep your root pointer private and hence the need for additional recursive methods which are private. 11. PART 2 In this part of the project, we will focus on testing and correctness. In our previous assignments, we ve all paid attention to making sure our programs execute correctly. The complexity of correctness grows as the complexity of our software increases. Pointers, which have been prevalent in most of our assignments, are a common source of additional errors. The binary search tree program features some methods which are quite complex in their number of special cases, and pointers figure prominently throughout this assignment. Thus we will take an additional step to ensure our software executes correctly. This part of the project asks us to create and implement a test plan. A test plan contains a number of test cases each of which tests the execution of some particular part of our code. A test plan may contain black box cases which assume the tester does not have access to the internal code; typically black box testing attempts to create all possible (or many) scenarios and to test the execution of the program in each of those scenarios. We will implement white box testing which assumes we have access to the code. Each test case in a white box test plan tests the correct execution of some branch of our code. Your job is to compose a series of white box test cases to test the execution of your program thoroughly. You will document each of your test cases by indicating such things as: what branch of code it tests, what is the input to the problem (tree structure), what is the expected output, what is the actual output of the program, and anything else you feel is important. An example test case is shown at the end of this assignment. 12. PART 3 For the third part of this project, we will be conducting an experiment in tree height. Add a method to your class that computes the height of the tree which is defined as the maximum number of branches from the root node to a leaf node. By definition, the root is at level 0 (height of 0). In this experiment, you will generate random integers to insert into your BST. After a certain number of insertions, you will measure the tree height. Conduct this experiment multiple times to compute an average height over a number of different insertion permutations. I recommend establishing several values for n, such as 25, 50, 75,.., Plot an average height point for each of these number of insertions. 3
4 Submit a graph with your results. Also include on the graph a curve for lg n so we can compare the average tree height with the ideal case. 13. Submissions: Please submit the following things in a tar file: Your implementation of the binary search tree in a file named bst.h. You can use the template on the class webpage as a starting point. Your testing plan as a pdf document. Your main program which computes the results of your experiment. A makefile which compiles your main program. A pdf of your graph showing your results. 4
5 Test Case 1 Method: remove remove(ptr) where ptr points to the root. Purpose: This case tests the special circumstance of removing the root node when it has 2 children. We want to be sure the root node is removed and reassigned to the root s successor node. Input: The test input for this case is a BST shown in the left side of Figure 1. It has a root node with value 20 and two children: left with 12 and right with 25. I will first use search() to assign a pointer ptr to point to the 20 node. I will use this ptr as input for the remove method: Node<T> *ptr = bst.search(20); bst.remove(ptr); Expected Output: The remove method is supposed to replace a node that has two children with the node s successor. In this test case, the successor node is the 25 node. It should replace the 20 root node as shown in the right side of Figure 1. Actual Output: The program correctly removed the 20 node and placed the 25 node as the new root. I can tell this is the structure because I altered the inorderwalk method to print the depth of each node along with its value. The 25 node reflects the new depth=0 correctly Figure 1: Example Tree for Test Case 1, remove(ptr>20) 5
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