Trees Types and Operations

1. TreesTypes and Operations

2. Agenda • General Trees • Binary Search Trees • AVL Trees • Heap Trees

3. 1. General Trees • Insertion • FIFO • LIFO • Key-sequenced Insertion • Deletion • Changing a General Tree into a Binary Tree

4. Insertion Given the parent Node a new node may be inserted as • FIFO

5. First in-first out (FIFO) insertion Data Structures: A Pseudocode Approach with C

6. Insertion Given the parent Node a new node may be inserted as • FIFO • LIFO

7. Last in-first out (LIFO) insertion Data Structures: A Pseudocode Approach with C

8. Insertion Given the parent Node a new node may be inserted as • FIFO • LIFO • Key-sequenced Insertion

9. Key-sequenced insertion Data Structures: A Pseudocode Approach with C

10. 1. General Trees • Insertion • FIFO • LIFO • Key-sequenced Insertion • Deletion • Changing a General Tree into a Binary Tree

11. Deletion • For general trees nodes to be deleted are restricted to be “leaves” • Otherwise a node maybe “purged”, i.e. a node is deleted along with all its children

12. 1. General Trees • Insertion • FIFO • LIFO • Key-sequenced Insertion • Deletion • Changing a General Tree into a Binary Tree

13. Changing into Binary Trees • Changing the meaning of the two pointers: • Leftchild …..first child • Rightchild ….. Next siblings

14. Changing a General Tree to a Binary Tree Data Structures: A Pseudocode Approach with C

15. Changing a General Tree to a Binary Tree Data Structures: A Pseudocode Approach with C

16. Changing a General Tree to a Binary Tree Data Structures: A Pseudocode Approach with C

17. Agenda • General Trees • Binary Search Trees • AVL Trees • Heap Trees

18. 2. Binary Search Tree • Basic Concepts • BST Operations • Threaded Trees

19. Basic Concepts • All items in left subtree < root • All items in right subtree > root

20. Binary Search Trees A binary search tree Not a binary search tree

21. Binary Search Tree Two binary search trees representing the same set:

22. 2. Binary Search Tree • Basic Concepts • BST Operations • Threaded Trees

23. BST Operations • Traversal • Search • Smallest ……….. ? • Largest …………? • Specific element • Insertion • Deletion

24. Inorder traversal of BST • Print out all the keys in sorted order Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

25. BST Operations • Traversal • Search • Smallest ……….. ? • Largest …………? • Specific element • Insertion • Deletion

26. findMin/ findMax • Return the node containing the smallest element in the tree • Start at the root and goes left/right as long as there is a left/right child. The stopping point is the smallest/largest element • Time complexity = O(height of the tree)

27. Searching BST (specific elem) • If we are searching for 15, then we are done. • If we are searching for a key < 15, then we should search in the left subtree. • If we are searching for a key > 15, then we should search in the right subtree.

28. Searching BST

29. BST Operations • Traversal • Search • Smallest ……….. ? • Largest …………? • Specific element • Insertion • Deletion

30. insert • Proceed down the tree as you would with a find • If X is found, do nothing (or update something) • Otherwise, insert X at the last spot on the path traversed • Time complexity = O(height of the tree)

31. BST Operations • Traversal • Search • Smallest ……….. ? • Largest …………? • Specific element • Insertion • Deletion

32. delete • When we delete a node, we need to consider how we take care of the children of the deleted node. • This has to be done such that the property of the search tree is maintained.

33. delete Three cases: (1) the node is a leaf • Delete it immediately (2) the node has one sub-tree (right or left) • Adjust a pointer from the parent to bypass that node

34. delete (3) the node has 2 children • replace the key of that node with the minimum element at the right subtree(or the maximum element at the left subtree) • delete the minimum element • Has either no child or only right child because if it has a left child, that left child would be smaller and would have been chosen. So invoke case 1 or 2. • Time complexity = O(height of the tree)

35. delete

36. 2. Binary Search Tree • Basic Concepts • BST Operations • Threaded Trees

37. Threaded Trees • Sparing recursion and stack • Making use of null right child of leaves to point to next node

38. Agenda • General Trees • Binary Search Trees • AVL Trees • Heap Trees

39. 3. AVL Trees • Properties • Operations

40. Properties of AVL Trees • It is a balanced binary tree(definition of Russian mathematicians Adelson-VelskiiandLandis) • The height of its sub-trees differs by no more than one (its balance factor is -1, 0, or 1), and its subtrees are also balanced.

41. Properties of AVL Trees • A sub tree is called • Left high (LH) if its balance is 1 • Equally high (EH) if it is 0 • Right high (RH) if it is -1

42. Operations on AVL Trees • Insertion and deletion are same as in BST • If unbalance occurs corresponding rotations must be performed to restore balance

43. Balanced trees: AVL tree rotations • Steps: • Check if case is case 1 or 2 of the following and act accordingly • Case 1: • tree is left high & • out-of-balance is created by a adding node to the left of the left sub-tree • …… One right rotation is needed Rotate out-of-balance node right

44. Case 1: single R-rotation • Case 1 * • Tree is left balanced • unbalance is caused by node on the left of left sub-tree h+2 h+1 h+1 h+1 h h h h h

45. Balanced trees: AVL tree rotations • Case 2: • tree is left high • out-of-balance is created by a adding node to the right of the left sub-tree • …… Two rotations are needed: Move from bottom of left sub-tree upwards till an unbalanced node is found and rotate it left Rotate left sub-tree right

46. Case 2: Double LR-rotation Add node to right of left balanced subtree h+2 h+1 h+2 h+2 h h h+1 h+1 h+1 h h h First rotation .. Left rotation of unbalanced node c Second rotation … Right rotation of left sub-tree g

47. End