Introduction to Data Structures.................

1. DATA STRUCTURES
DEF:DS is a collection of
organized data that are related to
each other.

2. DATA STRUCTURES
LINEAR DS NON-LINEAR
STACKS QUEUES LINKED LIST
TREES GRAPHS

3. TREES
 DEF:A TREE IS A SET OF NODES AND LINKS
 BINARY TREE:A BINARY TREE IS A FINITE SET OF ELEMENTS
THAT IS EITHER EMPTY OR PARTITIONED INTO 3 DISJOINT
SUBSETS .
 THE 1ST SUBSET CONTAINS A SINGLE ELEMENT CALLED AS
“ROOT OF THE BINARY TREE”.THE OTHER TWO SUBSETS ARE
THEMSELVES TREES CALLED LEFT AND RIGHT SUBTREES OF
THE ORIGINAL TREE.
 EACH ELEMENT OF THE BINARY TREE IS CALLED “NODE” OF
THE BINARY TREE.

4. A
B C
D F G
E
LEFT SUBTREE
RIGHT SUBTREE

5.  In the above example A is the root node of
the tree and B is the root of the left sub tree
and C is the root of the right subtree.
 Then A is called as father of B and C.
 Where B is the left son of A and C is the
right son of A.

6. TREE BASICS
 Number of nodes
 Height
 Root Node
 Leaves
 Interior nodes
 Number of levels
 Ancestors of H
 Descendants of B
 Siblings of E
 Right subtree

8. SIBLING
 If N is a node In T that has left sub tree S1,
and a right sub tree S2, then N is called as
the parent of the S1& S2
 So S1 &S2 are called siblings.

9. LEVEL NUMBER
 Every node in a tree is assigned a level
number.
 The root is defined at level 0
 The left and right child of root node has
level n0.1 and so on…………….

10. DEGREE
 The degree of the node is equal to the num
ber of children that a node has.
for eg: the degree of root node A is 2

11. IN DEGREE AND OUT DEGREE
 In in-degree of a node is the number of
edges arriving at that node.
for eg: the root node is the only node that
has an in-degree 0.
 Similarly the out degree of a node is the
number of edges that leaving the node.

12. Leaf(or)Leaf Node or Terminal node
 A “Node” that has no sons is called as “leaf
node”
 In the above fig D,E,F&G are the leaf nodes
 Where A,B,C are the non leaf nodes
A
B C
LEAF NODES

13. ANCESTOR
 A Node ‘n1’ is an ancestor of node ‘n2’ if
‘n1’ is neither father of ‘n2’(or) father of
some ancestor of ‘n2’. n1
n2
n1
B
n2

14.  In the above fig 1 ‘n1’ is the father of ‘n2’ so
directly we call ‘n1 as ancestor of ‘n2’
 In fig2 there is no direct relation ship b/w n1
and n2 even though n1 Is the father of B
where b is the father of n2 then we call n1
as father of ancestor of ‘n2’ so n1 become
ancestor of n2

15. Descendent
 A node ‘n2 is descendent of ‘n1’if ‘n2 is son
of ‘n1’ n1
 In the above fig directly n2 is the
n2
son of n1.
n1
In fig2 n2 is the son of ‘b’ where
b
B is the descendent of n1 the n2 is son of
descendent of n1 n2
Then n2 is descendent of n1.

16. Brother Nodes
 Two nodes are said to be brothers if they ae
left and right sons of the same father node
A
B C

17.  The direction of travelling from roots to
leaves is “down” and the travelling from
leaves to root is up.
 Moving from leaves is known as “climbing”
 And moving from roots to leaves is also
known as descending.

18. BINARY TREE
 A binary tree is a non-linear data structure
which is defined as collection of elements
called nodes.
 Every node has left pointer and right pointer
and the data element.
 Every binary tree has a root node which is
the top most node in the tree.
 If root is equal to null then the tree is empty.

19. 1
2 3
5 6 7
4
10 11 12
8
9

20.  In the above figure, node-2 is the left
successor and node-3 is the right succesor
 Note that the left sub-tree of the root node
consists of nodes-2,4,5,8,and 9. and right
sub-tree consists of nodes-3,6,7,10,11,12
 The leaf nodes are 5,8,9,10,11,12.

21. Properties of BinaryTree
 Property1:A tree with ‘n’ nodes has exactly
n-1 edges or branches.
 Property2:In a tree every node except the
root has exactly 1 parent
 Property3:There is exactly one path
connecting any 2 nodes in a tree.
 Property4:the max no of nodes in a binary
tree of height k is 2k+1 -1 (k>=0)
 Property5:Btree with n internal nodes has
n+1 external nodes.

22. Strict Binary Tree
 If every non leaf node in the binary tree
having non empty left and right sub trees
then it is known as strict binary trees
 In a binary tree if every non-leaf node
having both left and right sons then it is
called as strict binary tree.

23. A
B C
F G H
E
A
B C
D E
F

24.  In the above fig1 is a strict binary tree where
as fig2 is not because the non leaf node D
having only leftson ‘F’ it doesnot have right
son.
 So in a strict binary tree with n leaves totally
we have (2n-1) nodes.
 Ie from the above fig we have 4 leaf nodes
the n we have (2*4-1)=7

25. Level of the node:
 The level of the node in a binary tree is
defined as follows.
 The root of the binary tree always having
level ’0’ and the level of any other node is
one more that its father node.
 Level of a node=1+its father node level

26. LEVEL 0
A
LEVEL 1
B C
LEVEL 2 D E
LEVEL3 E

27. DEPTH OF BT
 The depth of bt is the max level of any leaf
node in that tree. A
LEVEL0
LEVEL1
B C
LEVEL2 D E
LEVEL3 F
G LEVEL4

28.  LEVEL:4
 DEPTH=4
 Depth always equals to length of longest
path from root to leaves

29. Complete BTree
 A complete Btree of Depth “D” is a strict
Btree whose all leaf nodes must exist at
level”D”.

30. REPRESENTATION OF B TREE
USING ARRAYS

31.  Suppose T is a Binary tree that is complete.
there is an efficient way of maintaining T
in memory called Sequential representation
of T.
The representation uses only a single linear
array tree as follows.
a)The root R of tree T is stored in Tree[i]
b)If a node N occupies Tree[K],then the left
child is stored in Tree[2k+1] and right child
is stored in Tree[2k+2].

32. 
Complete trees in arrays:
Storage of complete
Trees
0
1 2
3 4 5 6
7 8
…
0 1 2 3 4 5 6 7 8
k=3 2k+1, 2k+2
CIS 068

33. Sequential [1]
[2]
A
[3] B
Representation [4]
[5] C
(1) waste space
A [1] A [6]
(2) insertion/deletion
[2] B [7] D
problem
[3] -- [8]
B [4] [9] E
[5] C
A
[6] -- F
[7] -- H
C
[8]
[9] -- B I
C
D . D
[16] --
.
E D E F G
E
H I

34. REPRESENTATION OF B TREE
USING LINKED LIST

36. Data Structures: A
Pseudocode Approach
with C 36

37. Operations on BT
There are 3 important operations
on BT.
Insertion
Deletion
Searching