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C++ Programming: Program Design Including Data Structures, Fourth Edition

C++ Programming: Program Design Including Data Structures, Fourth Edition. Chapter 20: Binary Trees. Objectives. In this chapter, you will: Learn about binary trees Explore various binary tree traversal algorithms Learn how to organize data in a binary search tree

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C++ Programming: Program Design Including Data Structures, Fourth Edition

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  1. C++ Programming: Program Design Including Data Structures, Fourth Edition Chapter 20: Binary Trees

  2. Objectives In this chapter, you will: • Learn about binary trees • Explore various binary tree traversal algorithms • Learn how to organize data in a binary search tree • Learn how to insert and delete items in a binary search tree • Explore nonrecursive binary tree traversal algorithms C++ Programming: Program Design Including Data Structures, Fourth Edition

  3. Binary Trees C++ Programming: Program Design Including Data Structures, Fourth Edition

  4. Binary Trees (continued) Root node, and Parent of B and C Left child of A Right child of A Directed edge, directed branch, or branch Node Empty subtree (F’s right subtree) C++ Programming: Program Design Including Data Structures, Fourth Edition

  5. Binary Trees (continued) • Every node has at most two children • A node: • Stores its own information • Keeps track of its left subtree and right subtree • lLink and rLink pointers C++ Programming: Program Design Including Data Structures, Fourth Edition

  6. Binary Trees (continued) • A pointer to the root node of the binary tree is stored outside the tree in a pointer variable C++ Programming: Program Design Including Data Structures, Fourth Edition

  7. Binary Trees (continued) • Leaf: node that has no left and right children • U is parent of V if there’s a branch from U to V • There’s a unique path from root to every node • Length of a path: number of branches on path • Level of a node: number of branches on the path from the root to the node • The level of the root node of a binary tree is 0 • Height of a binary tree: number of nodes on the longest path from the root to a leaf C++ Programming: Program Design Including Data Structures, Fourth Edition

  8. A is the parent of B and C ABDG is a path (of length 3) from node A to node G A leaf • The longest path from root to a leaf is ABDGI • The number of nodes on this path is 5  the height of the tree is 5

  9. Binary Trees (continued) • How can we calculate the height of a binary tree? • This is a recursive algorithm: C++ Programming: Program Design Including Data Structures, Fourth Edition

  10. Copy Tree C++ Programming: Program Design Including Data Structures, Fourth Edition

  11. Binary Tree Traversal • Inorder traversal • Traverse the left subtree • Visit the node • Traverse the right subtree • Preorder traversal • Visit the node • Traverse the left subtree • Traverse the right subtree C++ Programming: Program Design Including Data Structures, Fourth Edition

  12. Binary Tree Traversal (continued) • Postorder traversal • Traverse the left subtree • Traverse the right subtree • Visit the node C++ Programming: Program Design Including Data Structures, Fourth Edition

  13. Binary Tree Traversal (continued) • Inorder sequence: listing of the nodes produced by the inorder traversal of the tree • Preorder sequence: listing of the nodes produced by the preorder traversal of the tree • Postorder sequence: listing of the nodes produced by the postorder traversal of the tree C++ Programming: Program Design Including Data Structures, Fourth Edition

  14. Binary Tree Traversal (continued) • Inorder sequence: B D A C • Preorder sequence: A B D C • Postorder sequence: D B C A C++ Programming: Program Design Including Data Structures, Fourth Edition

  15. Implementing Binary Trees • Typical operations: • Determine whether the binary tree is empty • Search the binary tree for a particular item • Insert an item in the binary tree • Delete an item from the binary tree • Find the height of the binary tree • Find the number of nodes in the binary tree • Find the number of leaves in the binary tree • Traverse the binary tree • Copy the binary tree C++ Programming: Program Design Including Data Structures, Fourth Edition

  16. Binary Search Trees • We can traverse the tree to determine whether 53 is in the binary tree  this is slow C++ Programming: Program Design Including Data Structures, Fourth Edition

  17. Binary Search Trees (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition

  18. Binary Search Trees (continued) • Every binary search tree is a binary tree C++ Programming: Program Design Including Data Structures, Fourth Edition

  19. Binary Search Trees (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition

  20. Search C++ Programming: Program Design Including Data Structures, Fourth Edition

  21. Insert C++ Programming: Program Design Including Data Structures, Fourth Edition

  22. Insert (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition

  23. Delete C++ Programming: Program Design Including Data Structures, Fourth Edition

  24. Delete (continued) • The delete operation has four cases: • The node to be deleted is a leaf • The node to be deleted has no left subtree • The node to be deleted has no right subtree • The node to be deleted has nonempty left and right subtrees C++ Programming: Program Design Including Data Structures, Fourth Edition

  25. Delete (continued) • To delete an item from the binary search tree, we must do the following: • Find the node containing the item (if any) to be deleted • Delete the node C++ Programming: Program Design Including Data Structures, Fourth Edition

  26. Binary Search Tree: Analysis • Let T be a binary search tree with n nodes, where n > 0 • Suppose that we want to determine whether an item, x, is in T • The performance of the search algorithm depends on the shape of T • In the worst case, T is linear C++ Programming: Program Design Including Data Structures, Fourth Edition

  27. Binary Search Tree: Analysis (continued) • Worst case behavior: T is linear • O(n) key comparisons C++ Programming: Program Design Including Data Structures, Fourth Edition

  28. Binary Search Tree: Analysis (continued) • Average-case behavior: • There are n! possible orderings of the keys • We assume that orderings are possible • S(n) and U(n): number of comparisons in average successful and unsuccessful case, respectively C++ Programming: Program Design Including Data Structures, Fourth Edition

  29. Binary Search Tree: Analysis (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition

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