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Chapter 11

Chapter 11. Binary Trees. Chapter Objectives. Learn about binary trees Explore various binary tree traversal algorithms Learn how to organize data in a binary search tree Discover how to insert and delete items in a binary search tree Explore nonrecursive binary tree traversal algorithms

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Chapter 11

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  1. Chapter 11 Binary Trees Data Structures Using C++

  2. Chapter Objectives • Learn about binary trees • Explore various binary tree traversal algorithms • Learn how to organize data in a binary search tree • Discover how to insert and delete items in a binary search tree • Explore nonrecursive binary tree traversal algorithms • Learn about AVL (height-balanced) trees Data Structures Using C++

  3. Binary Trees • Definition: A binary tree, T, is either empty or such that: • T has a special node called the root node; • T has two sets of nodes, LT and RT, called the left subtree and right subtree of T, respectively; • LT and RT are binary trees Data Structures Using C++

  4. Binary Tree Data Structures Using C++

  5. Binary Tree With One Node The root node of the binary tree = A LA = empty RA = empty Data Structures Using C++

  6. Binary Trees With Two Nodes Data Structures Using C++

  7. Binary Trees With Two Nodes Data Structures Using C++

  8. Various Binary Trees With Three Nodes Data Structures Using C++

  9. Binary Trees Following struct defines the node of a binary tree: template<class elemType> struct nodeType { elemType info; nodeType<elemType> *llink; nodeType<elemType> *rlink; }; Data Structures Using C++

  10. Nodes • For each node: • Data is stored in info • The pointer to the left child is stored in llink • The pointer to the right child is stored in rlink Data Structures Using C++

  11. General Binary Tree Data Structures Using C++

  12. Binary Tree Definitions • Leaf: node that has no left and right children • Parent: node with at least one child node • Level of a node: number of branches on the path from root to node • Height of a binary tree: number of nodes no the longest path from root to node Data Structures Using C++

  13. Height of a Binary Tree Recursive algorithm to find height of binary tree: (height(p) denotes height of binary tree with root p): if(p is NULL) height(p) = 0 else height(p) = 1 + max(height(p->llink), height(p->rlink)) Data Structures Using C++

  14. Height of a Binary Tree Function to implement above algorithm: template<class elemType> int height(nodeType<elemType> *p) { if(p == NULL) return 0; else return 1 + max(height(p->llink), height(p->rlink)); } Data Structures Using C++

  15. Copy Tree • Useful operation on binary trees is to make identical copy of binary tree • Use function copyTree when we overload assignment operator and implement copy constructor Data Structures Using C++

  16. Copy Tree template<class elemType> void copyTree(nodeType<elemType>* &copiedTreeRoot, nodeType<elemType>* otherTreeRoot) { if(otherTreeRoot == NULL) copiedTreeRoot = NULL; else { copiedTreeRoot = new nodeType<elemType>; copiedTreeRoot->info = otherTreeRoot->info; copyTree(copiedTreeRoot->llink, otherTreeRoot->llink); copyTree(copiedTreeRoot->rlink, otherTreeRoot->rlink); } }//end copyTree Data Structures Using C++

  17. Binary Tree Traversal • Must start with the root, then • Visit the node first or • Visit the subtrees first • Three different traversals • Inorder • Preorder • Postorder Data Structures Using C++

  18. Traversals • Inorder • Traverse the left subtree • Visit the node • Traverse the right subtree • Preorder • Visit the node • Traverse the left subtree • Traverse the right subtree Data Structures Using C++

  19. Traversals • Postorder • Traverse the left subtree • Traverse the right subtree • Visit the node Data Structures Using C++

  20. Binary Tree: Inorder Traversal Data Structures Using C++

  21. Binary Tree: Inorder Traversal template<class elemType> void inorder(nodeType<elemType> *p) { if(p != NULL) { inorder(p->llink); cout<<p->info<<” “; inorder(p->rlink); } } Data Structures Using C++

  22. Binary Tree: Traversals template<class elemType> void postorder(nodeType<elemType> *p) { if(p != NULL) { postorder(p->llink); postorder(p->rlink); cout<<p->info<<” “; } }1 template<class elemType> void preorder(nodeType<elemType> *p) { if(p != NULL) { cout<<p->info<<” “; preorder(p->llink); preorder(p->rlink); } } Data Structures Using C++

  23. Public isEmpty inorderTraversal preorderTraversal postorderTraversal treeHeight treeNodeCount treeLeavesCount destroyTree Implementing Binary Trees: class binaryTreeType Functions • Private • copyTree • Destroy • Inorder, preorder, postorder • Height • Max • nodeCount • leavesCount Data Structures Using C++

  24. Binary Search Trees • Data in each node • Larger than the data in its left child • Smaller than the data in its right child • A binary search tree,t, is either empty or: • T has a special node called the root node • T has two sets of nodes, LT and RT, called the left subtree and right subtree of T, respectively • Key in root node larger than every key in left subtree and smaller than every key in right subtree • LT and RT are binary search trees Data Structures Using C++

  25. Binary Search Trees Data Structures Using C++

  26. Operations Performed on Binary Search Trees • Determine whether the binary search tree is empty • Search the binary search tree for a particular item • Insert an item in the binary search tree • Delete an item from the binary search tree Data Structures Using C++

  27. Operations Performed on Binary Search Trees • Find the height of the binary search tree • Find the number of nodes in the binary search tree • Find the number of leaves in the binary search tree • Traverse the binary search tree • Copy the binary search tree Data Structures Using C++

  28. Binary Search Tree Analysis Worst Case: Linear tree Data Structures Using C++

  29. Binary Search Tree Analysis • Theorem: Let T be a binary search tree with n nodes, where n > 0.The average number of nodes visited in a search of T is approximately 1.39log2n • Number of comparisons required to determine whether x is in T is one more than the number of comparisons required to insert x in T • Number of comparisons required to insert x in T same as the number of comparisons made in unsuccessful search, reflecting that x is not in T Data Structures Using C++

  30. Binary Search Tree Analysis It follows that: It is also known that: Solving Equations (11-1) and (11-2) Data Structures Using C++

  31. Nonrecursive Inorder Traversal Data Structures Using C++

  32. Nonrecursive Inorder Traversal: General Algorithm • current = root; //start traversing the binary tree at // the root node • while(current is not NULL or stack is nonempty) if(current is not NULL) { push current onto stack; current = current->llink; } else { pop stack into current; visit current; //visit the node current = current->rlink; //move to the //right child } Data Structures Using C++

  33. Nonrecursive Preorder Traversal: General Algorithm 1. current = root; //start the traversal at the root node 2. while(current is not NULL or stack is nonempty) if(current is not NULL) { visit current; push current onto stack; current = current->llink; } else { pop stack into current; current = current->rlink; //prepare to visit //the right subtree } Data Structures Using C++

  34. Nonrecursive Postorder Traversal • current = root; //start traversal at root node • v = 0; • if(current is NULL) the binary tree is empty • if(current is not NULL) • push current into stack; • push 1 onto stack; • current = current->llink; • while(stack is not empty) if(current is not NULL and v is 0) { push current and 1 onto stack; current = current->llink; } Data Structures Using C++

  35. Nonrecursive Postorder Traversal (Continued) else { pop stack into current and v; if(v == 1) { push current and 2 onto stack; current = current->rlink; v = 0; } else visit current; } Data Structures Using C++

  36. AVL (Height-balanced Trees) • A perfectly balanced binary tree is a binary tree such that: • The height of the left and right subtrees of the root are equal • The left and right subtrees of the root are perfectly balanced binary trees Data Structures Using C++

  37. Perfectly Balanced Binary Tree Data Structures Using C++

  38. AVL (Height-balanced Trees) • An AVL tree (or height-balanced tree) is a binary search tree such that: • The height of the left and right subtrees of the root differ by at most 1 • The left and right subtrees of the root are AVL trees Data Structures Using C++

  39. AVL Trees Data Structures Using C++

  40. Non-AVL Trees Data Structures Using C++

  41. Insertion Into AVL Tree Data Structures Using C++

  42. Insertion Into AVL Trees Data Structures Using C++

  43. Insertion Into AVL Trees Data Structures Using C++

  44. Insertion Into AVL Trees Data Structures Using C++

  45. Insertion Into AVL Trees Data Structures Using C++

  46. AVL Tree Rotations • Reconstruction procedure: rotating tree • left rotation and right rotation • Suppose that the rotation occurs at node x • Left rotation: certain nodes from the right subtree of x move to its left subtree; the root of the right subtree of x becomes the new root of the reconstructed subtree • Right rotation at x: certain nodes from the left subtree of x move to its right subtree; the root of the left subtree of x becomes the new root of the reconstructed subtree Data Structures Using C++

  47. AVL Tree Rotations Data Structures Using C++

  48. AVL Tree Rotations Data Structures Using C++

  49. AVL Tree Rotations Data Structures Using C++

  50. AVL Tree Rotations Data Structures Using C++

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