Common Use of Tree as a Data Structure ADVANCE 1. Nodes 2. Parent Nodes & Child Nodes 3. Leaf Nodes 4. Root Node 5. Sub Tree 6. Level of a tree: 7. m-ary Tree 8. Binary Tree (BT) 9. Complete and Full Binary Tree 10. Traversal 11. Binary Search Tree (BST) 12. Inorder Traversal – Left_ParentNode_Right 13. Postorder Traversal – Left_Right_ParentNode 14. Preorder Traversal – ParentNode_Left_Right 15. Binary Search Tree (BST) 16. BST - Insert, Delete

1. DATA STRUCTURE
TREE

2. DATA STRUCTURE - TREE
 Many applications are
hierarchical in nature.
 Linear data structures are not
appropriate for these type of
applications.
Tree  2
President
Vice President
Executive Executive
Vice President
Executive
Vice President
Executive Executive
Mashrafi Khan
Mairufa Khan
Junayed
Ahmed
Masrufa
Ahmed
Fahim Khan
Ashfar
Khan
Moin Khan
Imran
Khan
Bushra
Khan
Faizul Khan
Mahboob
Khan
Mashrufa
Khan
Mashraba
Khan
Hierarchical structure of a company
Genealogy tree of a family

3. COMMON USE OF TREE AS A DATA STRUCTURE
 Representing hierarchical data
 Storing data in a way that makes it easily searchable
 Representing sorted lists of data
 As a workflow for compositing digital images for visual effects
 Routing algorithms
Tree  3

4. TREE - DEFINITION
 As a data structure, a tree consists of one
or more nodes, where each node has a
value and a list of references to other (its
children) nodes.
 A tree must have a node designated as
root. If a tree has only one node, that
node is the root node. Root is never
referenced by any other node.
 Having more than one node indicates that
the root have some (at least one)
references to its children nodes and the
children nodes (might) have references to
their children nodes and so on.
Tree  4
A
B C D
E F JIHG
K ML ON
ROOT
VALUE
CHILDCHILD

5. TREE - DEFINITION
 Generally it is considered that in a tree,
there is only one path going from one
node to another node.
 There cannot be any cycle or loop.
 The link from a node to other node is
called an edge.
 An arrowed edge
indicates flow from P to Q.
 An straight line edge
indicates flow from P to Q and Q to P.
Tree  5
A
B C D
E F JIHG
K ML ON
If node F is reached through node B, than the link from
node K to node F will not be considered.
If link from node L to node C is considered, than there
will be a cycle among nodes C, G, and L.
P Q
P Q
A straight line is generally used to represent the links
between the nodes of a tree.

6. TREE - DEFINITION
 Nodes
 Parent Nodes & Child Nodes
 Leaf Nodes: nodes with no child
 Root Node: node with no parent
 Sub Tree: the tree rooted by a child
 Level of a tree:
 Root at level 0;
 Each children have the level one more than
its parent.
 Height/depth of the tree: Total
number of Levels
 Height of a node: Total number of
levels from bottom
[Tree height – node level].
Tree  6
A
B C D
E F JIHG
K ML ON
LEVEL
0
1
2
3
Height of this tree is 4, as there are four levels (0…3).
Height of root A is 4;
Height of nodes B, C, D is 3;
Height of E, F, G, H, I, J is 2;
Height of nodes K, L , M, N, O is 1.

7. m-ARY TREE
 A Tree is an m-ary Tree when each of its node has no more than m children.
Tree  7
A
B C D
E F JIHG
K ML ON
A
B D
E F IH
GK
ML
ON
2-ary tree 3-ary tree

8. BINARY TREE (BT)
 Each node of a binary Tree has at most 2 children.
Tree  8
A
B D
E F IH
GK
ML
ON
Binary tree

9. COMPLETE AND FULL BINARY TREE
 A complete/full binary tree is a binary tree,
which is completely filled with nodes from
top to bottom and left to right.
 The tree starts from the root (top), goes to
the next level with first the left child and
then the right child (left to right). The
process repeats for each next level with
each node till the last (bottom) level.
 In complete binary tree some of the nodes
only for the bottom level might be absent
(here nodes after N).
 In Full binary tree the last/bottom level
must also be filled up.
Tree  9
A
B D
E F IH
GK ML N C J P

10. COMPLETE AND FULL BINARY TREE
 A full binary tree of depth n is a binary tree of depth n with 2n - 1 nodes, n >=0.
 A binary tree with k nodes and depth n is a complete binary tree if and only if its
nodes correspond to the nodes numbered from 0 to k-1 in the full binary tree of
depth n.
Tree  10
A
B D
E F IH
GK ML N C J P
Level=0,
# of nodes=20=1
Level=1,
# of nodes=21=2
Level=2,
# of nodes=22=4
Level=3,
# of nodes=23=8
Total Number of Nodes:
20+ 21+ 22+ 23 = 24 - 1 = 15
Height of the tree: Log215 = 4
If total number of nodes are n,
Nodes at each level L = 2L
Bottom Level = BL
Height h = BL+1 = Log2n
Total nodes = 2h – 1 = n

11. TRAVERSAL
 Systematic way of visiting all the nodes.
 Methods:
 Inorder
 Postorder
 Preorder
 They all traverse the left subtree before the right subtree.
Tree  11

12. INORDER TRAVERSAL – LEFT_PARENTNODE_RIGHT
 Traverse the left subtree
 Visit the parent node
 Traverse the right subtree
Tree  12
A
B D
E F H
K ML C
K E L B F M A H C D

13. POSTORDER TRAVERSAL – LEFT_RIGHT_PARENTNODE
 Traverse the left subtree
 Traverse the right subtree
 Visit the parent node
Tree  13
A
B D
E F H
K ML C
K L E M F B C H D A

14. PREORDER TRAVERSAL – PARENTNODE_LEFT_RIGHT
 Visit the parent node
 Traverse the left subtree
 Traverse the right subtree
Tree  14
A
B D
E F H
K ML C
A B E K L F M D H C

15. BINARY SEARCH TREE (BST)
 Is a Binary Tree such that:
 Every node entry has a unique key (i.e. no duplication item).
 All the keys in the left subtree of a node are less than the key of the node.
 All the keys in the right subtree of a node are greater than the key of the node.
Tree  15
43
31 64
20 40 8956
3328 47 59
Fred
Dan Mary
Alan Eve SueKate
EricBill Greg Len
Integer Key String Key

16. BST - INSERT
Tree  16
43
31 64
20 40 8956
3328 47 59
> <
< <
<><
>
>
>
Fred
Dan Mary
Alan Eve SueKate
EricBill Greg Len
> <
< >
> <<
>
>
<
Integer Key String Key
43 31 64 40 20 89 56 47 33 28 59Fred Mary Kate Dan Len Alan Eve Bill Sue Greg Eric

17. BST - SEARCH
 Search Elements 59 and 42
Fahad Ahmed CSC 2015: Data Structures Tree  17
43
31 64
20 40 8956
3328 47 59
43 < 59
64 > 59
56 < 59
59 = 59
43 > 42
31 < 42
40 < 42
? 42

18. BST - DELETE
 Delete 47 (leaf node);
 Delete 40 (have only one child);
 Delete 64 and 31 (have both child).
Tree  18
43
31 64
20 40 8956
3828 47 59
32
33
5932
Leaf Node: Just DeleteLeaf Node: Just Delete
Node with one child:
connect the parent to
the child and Delete
Node with one child: connect the
parent to the child and Delete
Node with two
children on right
subtree: Replace with
the child with highest
value either from left
or right subtree
Node with two
children on left
subtree: Replace with
the child with lowest
value either from left
or right subtree