1. Trees Types and Operations

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

3. 1. General Trees Insertion Deletion
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 Deletion
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 Deletion
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 Insertion Deletion 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 Insertion Deletion 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 Insertion Deletion Smallest ……….. ?
Largest …………?
Specific element
Insertion
Deletion

30. insert Time complexity = O(height of the tree)
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 Insertion Deletion 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-Velskii and Landis)
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
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