This document discusses various non-linear data structures including binary trees, binary search trees, and AVL trees. It provides definitions and examples of key tree concepts like nodes, leaves, paths, levels, and traversals. It also describes routines for operations on binary search trees like insertion, finding minimum/maximum, and deletion. For AVL trees, it explains how they differ from regular BSTs by restricting the height difference between subtrees to 1, and how rotations are used to rebalance the tree during insertions that cause imbalances.

1. CS1301 Data Structures

2. Unit III Non LINEAR DATA STRUCTURE
 Preliminaries
 Binary Tree
 The Search Tree ADT
 Binary Search Tree
 AVL Tree
 Tree Traversals
 Hashing - General Idea
 Hash Function
 Separate Chaining
 OpenAddressing
 Linear Probing
 Priority Queue (Heaps) - Model - Simple Implementations
 Binary Heap

3. Trees
• Finite set of one or more nodes that there is a
specially designated node called root and zero or
more non empty sub trees T1, T2,.. each of whose
roots are connected by a directed edge from Root
R
• A tree T is a set of nodes storing elements such
that the nodes have a parent-child relationship
that satisfies the following
– if T is not empty, T has a special tree called the root
that has no parent
– each node v of T different than the root has a unique
parent node w; each node with parent w is a child of w

4. Trees

5. Trees
• Root : node which doesn’t have a parent
• Node : Item of information
• Leaf/ terminal : node which doesn’t have children
• Siblings : children of same parents
• Path : sequence of nodes n1,n2,n3,….,nk such
that ni is parent of ni+1 for 1<= i<k. there is
exactly only one path from each node to root;
path from A to L is A,C,F,L
• Length : no. of edges on the path ; length of A to
L is 3
• Degree : number of subtrees of a node; A -> 4, C-
>2 degree of tree is maximum no.of any node in
the tree ; degree of tree is 4

6. Trees
• Level : initially letting the root be at level one,
if a node is at level L then its children are at
level L + 1
– Level of A is 1; Level of B, C, D is 2; Level of
F,G,H,I,J is 3, Level of K,L,M is 4
• Depth : For any node n, the depth of n is the
unique path from root to n
– Depth of root is 0; Depth of L is 3
• Height : For any node n, the height of the node
is the length of the longest path from n to the
leaf

7. CHAPTER 5 7
Level and Depth
K L
E F
B
G
C
M
H I J
D
A
Level
1
2
3
4
node (13)
degree of a node
leaf (terminal)
nonterminal
parent
children
sibling
degree of a tree (3)
ancestor
level of a node
3
2 1 3
2 0 0 1 0 0
0 0 0
1
2 2 2
3 3 3 3 3 3
4 4 4

8. Trees
• Binary Tree
–Tree in which no node can have more
than two children

9. Trees
• Binary Tree Node Declaration
Struct Treenode
{
int element;
Struct Treenode *Left;
Struct TreeNode *Right;
};

10. Trees
• Binary Tree
–Comparison between general tree &
Binary Tree
General Tree Binary Tree
Has any no. of children Has not more than two
children

11. Trees
• Full Binary Tree
– Full binary tree of h has 2^h+1 -1 nodes

12. Trees
• Complete Binary Tree
– A complete binary tree is a binary tree, which is
completely filled, with the possible exception of
the bottom level, which is filled from left to right
– A full binary tree can be a complete binary tree,
but all complete binary tree is not a full binary tree

13. Trees
• Representation of Binary Tree
–Two ways
• Linear Representation
• Linked Representation

14. Trees
• Linear Representation of Binary
Tree
– Elements are represented using arrays
– For any element in position i, the left child is in
position 2i, right child is in position (2i+1) and
parent in position (i/2)

15. Trees
• Linked Representation of Binary
Tree
– Elements are represented using pointers
– Each node in linked representation has two fields
• Pointer to left subtree
• Data field
• Pointer to right subtree
– In leaf node, both pointer fields are assigned as
NULL

16. Trees
• Expression Tree
– Is a binary tree
– Leaf nodes are operands and interior nodes are
operators
– Expression tree be traversed by inorder , preorder,
postorder traversal

17. Trees
• Constructing an Expression Tree
1. Read one symbol at a time from postfix
expression
2. Check whether symbol is an operand or
operator
1. If operand, create a one –node tree and
push a pointer on to the stack
2. If operator, pop two pointers from the
stack namely T1 and T2 and form a new
tree with root as the operator and T2 as a
left child and T1 as a right child. A
pointer to this new tree is pushed onto the
stack

18. Trees
• Example

19. Trees
• Example

20. Trees
• Binary Search Tree
–Is a binary tree
–For every node x in the tree, value of all
the keys in its left subtree are smaller than
the value X and value of all the keys in the
right subtree are larger than the value X

21. Trees
• Binary Search Tree
–Every binary search tree is a binary tree
–All binary tree need not be bnary
search tree

22. Trees
• Declaration Routine for Binary Search
Tree

23. Trees
• Routine for Make an empty tree

24. Trees
• Insertion

25. Trees
• Insertion - Example

26. Trees
• Insertion - Example

27. Trees
• Routine for Insertion

28. Trees
• Find

29. Trees
• Find – Example to find 10

30. Trees
• Find – Example to find 10

31. Trees
• Routine for Find

32. Trees
• Find Minimum
–Returns position of the smallest element in
a tree
–Start at the root and go left as long as there
is a left child, the stopping point will be
the smallest element

33. Trees
• Find Minimum – example

34. Trees
• Find Minimum – Recursive routine

35. Trees
• Find Minimum – Non Recursive
routine

36. Trees
• Find Maximum
–Returns the largest element in the tree
–To perform, start at the root and go right as
long as there is a right child
–Stopping point is the largest element

37. Trees
• Find Maximum - Example

38. Trees
• Find Maximum – Recursive Routine

39. Trees
• Find Maximum – Non Recursive
Routine

40. Trees
• Deletion
–Complex in BST
–To delete an element, consider the
following 3 possibilities
• Node to be deleted is a leaf node
• Node with one child
• Node with two children

41. Trees
• Deletion
– Node to be deleted is a leaf node

42. Trees
• Deletion
• Node with one child : deleted by adjusting its
parent pointer that points to its child node

43. Trees
• Deletion
• Node with two children : to replace the data of
node to be deleted with its smallest data of right subtree
and recursively delete that node

44. Trees
• Deletion
• Node with two children

45. Trees
• Deletion
• Node with two children

46. Trees
• Deletion
• Node with two children

47. Trees
• Deletion
• Node with two children

48. Trees
• Deletion
• Node with two children

49. Trees
• Deletion
• Node with two children

50. Trees
• Deletion Routine

51. Trees
• Deletion Routine

52. Trees
• AVL Tree (Adelson - Velskill and
Landis)
–Binary search tree except that for every
node in the tree, the height of left and right
subtrees can differ by atmost 1
–Balance factor is the height of left subtree
minus height of right subtree; -1, 0, or +1
–If Balance factor is less than or greater than
1, the tree has to be balanced by making
single or double rotations

53. Trees
• AVL Tree (Adelson Velskill and
Landis)

54. Trees
• AVL Tree (Adelson Velskill and
Landis)

55. Trees
• AVL Tree (Adelson Velskill and Landis)
– AVL tree causes imbalance, when anyone of the
following conditions occur
• An insertion into the left subtree of the left child of
node
• An insertion into the right subtree of the left child
of node
• An insertion into the left subtree of the right child
of node
• An insertion into the right subtree of the right
child of node
Imbalances can overcome by
1. Single rotation
2. Double rotation

56. Trees
• AVL Tree (Adelson Velskill and
Landis)
• Single rotation
– Performed to fix case1 and case 4
Case 1: An insertion into the left subtree of the left
child of node

57. Trees
• AVL Tree (Adelson Velskill and
Landis)
• Single rotation
– Performed to fix case1 and case 4
Case 1: An insertion into the left subtree of the left
child of node