1 Nell Dale Chapter 8 Binary Search Trees Modified from the slides by Sylvia Sorkin, Community College of Baltimore County - Essex Campus C++ Plus Data Structures

2 for an element in a sorted list stored sequentially l in an array: O(Log 2 N) l in a linked list: ? (midpoint = ?) Binary search

3 l Introduce some basic tree vocabulary l Develop algorithms l Implement operations needed to use a binary search tree Goals of this chapter

4 Jake’s Pizza Shop Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len

5 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len A Tree Has a Root Node ROOT NODE

6 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len Leaf nodes have no children LEAF NODES

7 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len A Tree Has Levels LEVEL 0 Level of a node: its distance from the root

8 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len Level One LEVEL 1

9 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len Level Two LEVEL 2

10 l Maximum number of levels: N l Minimum number of levels: logN + 1 e.g., N=8, 16, …, 1000 A binary tree with N nodes:

11 the maximum level in a tree The height of a tree: -- a critical factor in determining how efficiently we can search for elements

12 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len A Subtree LEFT SUBTREE OF ROOT NODE

13 Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len Another Subtree RIGHT SUBTREE OF ROOT NODE

14 A binary tree is a structure in which: Each node can have at most two children, and in which a unique path exists from the root to every other node. The two children of a node are called the left child and the right child, if they exist. Binary Tree

15 A Binary Tree Q V T K S A E L

16 A Binary Tree ? Q V T K S A E L

17 How many leaf nodes? Q V T K S A E L

18 How many descendants of Q? Q V T K S A E L

19 How many ancestors of K? Q V T K S A E L

20 Implementing a Binary Tree with Pointers and Dynamic Data Q V T K S A E L

21 Each node contains two pointers template struct TreeNode { ItemType info; // Data member TreeNode * left; // Pointer to left child TreeNode * right; // Pointer to right child };. left. info. right NULL ‘A’ 6000

// BINARY SEARCH TREE SPECIFICATION template class TreeType { public: TreeType ( ); // constructor ~TreeType ( ); // destructor bool IsEmpty ( ) const; bool IsFull ( ) const; int NumberOfNodes ( ) const; void InsertItem ( ItemType item ); void DeleteItem (ItemType item ); void RetrieveItem ( ItemType& item, bool& found ); void PrintTree (ofstream& outFile) const;... private: TreeNode * root; }; 22

23 TreeType CharBST; ‘J’ ‘E’ ‘A’ ‘S’ ‘H’ TreeType ~TreeType IsEmpty InsertItem Private data: root RetrieveItem PrintTree.

24 A Binary Tree Q V T K S A E L Search for ‘S’?

25 A special kind of binary tree in which: 1. Each node contains a distinct data value, 2. The key values in the tree can be compared using “greater than” and “less than”, and 3. The key value of each node in the tree is less than every key value in its right subtree, and greater than every key value in its left subtree. A Binary Search Tree (BST) is...

26 Depends on its key values and their order of insertion. Insert the elements ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ in that order. The first value to be inserted is put into the root node. Shape of a binary search tree... ‘J’

27 Thereafter, each value to be inserted begins by comparing itself to the value in the root node, moving left it is less, or moving right if it is greater. This continues at each level until it can be inserted as a new leaf. Inserting ‘E’ into the BST ‘J’ ‘E’

28 Begin by comparing ‘F’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf. Inserting ‘F’ into the BST ‘J’ ‘E’ ‘F’

29 Begin by comparing ‘T’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf. Inserting ‘T’ into the BST ‘J’ ‘E’ ‘F’ ‘T’

30 Begin by comparing ‘A’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf. Inserting ‘A’ into the BST ‘J’ ‘E’ ‘F’ ‘T’ ‘A’

31 is obtained by inserting the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order? What binary search tree... ‘A’

32 obtained by inserting the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order. Binary search tree... ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ A “degenerate” tree!

33 Another binary search tree Add nodes containing these values in this order: ‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’ ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’ ‘K’ ‘P’

34 Is ‘F’ in the binary search tree? ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’ ‘K’ ‘V’ ‘P’ ‘Z’‘D’‘Q’‘L’‘B’‘S’

// BINARY SEARCH TREE SPECIFICATION template class TreeType { public: TreeType ( ) ; // constructor ~TreeType ( ) ; // destructor bool IsEmpty ( ) const ; bool IsFull ( ) const ; int NumberOfNodes ( ) const ; void InsertItem ( ItemType item ) ; void DeleteItem (ItemType item ) ; void RetrieveItem ( ItemType& item, bool& found ) ; void PrintTree (ofstream& outFile) const ;... private: TreeNode * root ; }; 35

// SPECIFICATION (continued) // // RECURSIVE PARTNERS OF MEMBER FUNCTIONS template void PrintHelper ( TreeNode * ptr, ofstream& outFile ) ; template void InsertHelper ( TreeNode * & ptr, ItemType item ) ; template void RetrieveHelper ( TreeNode * ptr, ItemType& item, bool& found ) ; template void DestroyHelper ( TreeNode * ptr ) ; 36

// BINARY SEARCH TREE IMPLEMENTATION // OF MEMBER FUNCTIONS AND THEIR HELPER FUNCTIONS // template TreeType :: TreeType ( ) // constructor { root = NULL ; } // template bool TreeType :: IsEmpty( ) const { return ( root == NULL ) ; } 37

template void TreeType :: RetrieveItem ( ItemType& item, bool& found ) { RetrieveHelper ( root, item, found ) ; } template void RetrieveHelper ( TreeNode * ptr, ItemType& item, bool& found) { if ( ptr == NULL ) found = false ; else if ( item info )// GO LEFT RetrieveHelper( ptr->left, item, found ) ; else if ( item > ptr->info ) // GO RIGHT RetrieveHelper( ptr->right, item, found ) ; else { // item = ptr->info ; found = true ; } 38

template void TreeType :: RetrieveItem ( ItemType& item, KF key, bool& found ) // Searches for an element with the same key as item’s key. // If found, store it into item { RetrieveHelper ( root, item, key, found ) ; } template void RetrieveHelper ( TreeNode * ptr, ItemType& item, KF key, bool& found) { if ( ptr == NULL ) found = false ; else if ( key info.key() )// GO LEFT RetrieveHelper( ptr->left, item, key, found ) ; else if ( key > ptr->info.key() ) // GO RIGHT RetrieveHelper( ptr->right, item, key, found ) ; else { item = ptr->info ; found = true ; } 39

template void TreeType :: InsertItem ( ItemType item ) { InsertHelper ( root, item ) ; } template void InsertHelper ( TreeNode * & ptr, ItemType item ) { if ( ptr == NULL ) { // INSERT item HERE AS LEAF ptr = new TreeNode ; ptr->right = NULL ; ptr->left = NULL ; ptr->info = item ; } else if ( item info )// GO LEFT InsertHelper( ptr->left, item ) ; else if ( item > ptr->info ) // GO RIGHT InsertHelper( ptr->right, item ) ; } 40

41 l Find the node in the tree l Delete the node from the tree l Three cases: 1. Deleting a leaf 2. Deleting a node with only one child 3. Deleting a node with two children Note: the tree must remain a binary tree and the search property must remain intact Delete()

template void TreeType :: DeleteItem ( ItemType& item ) { DeleteHelper ( root, item ) ; } template void DeleteHelper ( TreeNode * ptr, ItemType& item) { if ( item info )// GO LEFT DeleteHelper( ptr->left, item ) ; else if ( item > ptr->info ) // GO RIGHT DeleteHelper( ptr->right, item ) ; else { DeleteNode(ptr); // Node is found: call DeleteNode } 42

template void DeleteNode ( TreeNode *& tree ) { ItemType data; TreeNode * tempPtr; tempPtr = tree; if ( tree->left == NULL ) { tree = tree->right; delete tempPtr; } else if (tree->right == NULL ) { tree = tree->left; delete tempPtr; } else { // have two children GetPredecessor(tree->left, item); tree->info = item; DeleteHelper(tree->left, item); // Delete predecessor node (rec) } 43

template void GetPredecessor( TreeNode * tree, ItemType& data ) // Sets data to the info member of the rightmost node in tree { while (tree->right != NULL) tree = tree->right; data = tree->info; } 44

45 Traverse a list: -- forward -- backward Traverse a tree: -- there are many ways! PrintTree()

46 Inorder Traversal: A E H J M T Y ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’‘Y’ tree Print left subtree firstPrint right subtree last Print second

// INORDER TRAVERSAL template void TreeType :: PrintTree ( ofstream& outFile ) const { PrintHelper ( root, outFile ) ; } template void PrintHelper ( TreeNode * ptr, ofstream& outFile ) { if ( ptr != NULL ) { PrintHelper( ptr->left, outFile ) ;// Print left subtree outFile info ; PrintHelper( ptr->right, outFile ) ;// Print right subtree } 47

48 Preorder Traversal: J E A H T M Y ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’‘Y’ tree Print left subtree secondPrint right subtree last Print first

49 ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’‘Y’ tree Print left subtree firstPrint right subtree second Print last Postorder Traversal: A H E M Y T J

template TreeType :: ~TreeType ( )// DESTRUCTOR { DestroyHelper ( root ) ; } template void DestroyHelper ( TreeNode * ptr ) // Post: All nodes of the tree pointed to by ptr are deallocated. { if ( ptr != NULL ) { DestroyHelper ( ptr->left ) ; DestroyHelper ( ptr->right ) ; delete ptr ; } 50

51 Read Text … Iterative Insertion and Deletion

52 l Is the depth of recursion relatively shallow? Yes. l Is the recursive solution shorter or clearer than the nonrecursive version? Yes. l Is the recursive version much less efficient than the nonrecursive version? No. Recursion or Iteration? Assume: the tree is well balanced.

53 Use a recursive solution when (Chpt. 7): The depth of recursive calls is relatively “shallow” compared to the size of the problem. l The recursive version does about the same amount of work as the nonrecursive version. l The recursive version is shorter and simpler than the nonrecursive solution. SHALLOW DEPTH EFFICIENCY CLARITY

54 BST: l Quick random-access with the flexibility of a linked structure l Can be implemented elegantly and concisely using recursion l Takes up more memory space than a singly linked list l Algorithms are more complicated Binary Search Trees (BSTs) vs. Linear Lists

55 End