1 / 57

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

laraine
Download Presentation

Chapter 11

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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++

More Related