Week 8 (trees)

Data Structures
Tree
Binary Tree
Binary Search Tree
Operations

Tree Data Structures
• There are a number of applications where
linear data structures are not appropriate.
• Consider a genealogy tree of a family.
Mohammad Aslam Khan
Sohail Aslam Javed Aslam Yasmeen Aslam
SaadHaaris Qasim Asim Fahd Ahmad Sara Omer

Tree Data Structure
• A linear linked list will not be able to
capture the tree-like relationship with
ease.
• Shortly, we will see that for applications
that require searching, linear data
structures are not suitable.
• We will focus our attention on binary trees.

Binary Tree
• A binary tree is a finite set of elements that is
either empty or is partitioned into three disjoint
subsets.
• The first subset contains a single element called
the root of the tree.
• The other two subsets are themselves binary
trees called the left and right subtrees.
• Each element of a binary tree is called a node of
the tree.

Binary Tree
• Binary tree with 9 nodes.
A
B
D
H
C
E F
G I

Binary Tree
A
B
D
H
C
E F
G I
Left subtree
root
Right subtree

Binary Tree
• Recursive definition
A
B
D
H
C
E F
G ILeft subtree
root
Right subtree

Binary Tree
A
B
D
H
C
E F
G I
Left subtree
root

Binary Tree
A
B
D
H
C
E F
G I
root

Binary Tree
A
B
D
H
C
E F
G I
root
Right subtree

Binary Tree
A
B
D
H
C
E F
G I
root
Right subtreeLeft subtree

Not a Tree
• Structures that are not trees.
A
B
D
H
C
E F
G I

Binary Tree: Terminology
A
B
D
H
C
E F
G I
parent
Left descendant Right descendant
Leaf nodes Leaf nodes

Binary Tree
• If every non-leaf node in a binary tree has non-empty left
and right subtrees, the tree is termed a strictly binary
tree.
A
B
D
H
C
E F
G I
J
K

Level of a Binary Tree Node
• The level of a node in a binary tree is
defined as follows:
 Root has level 0,
 Level of any other node is one more than the
level its parent (father).
• The depth of a binary tree is the maximum
level of any leaf in the tree.

Level of a Binary Tree Node
A
B
D
H
C
E F
G I
1
0
1
2 2 2
3 3 3
Level 0
Level 1
Level 2
Level 3

Complete Binary Tree
• A complete binary tree of depth d is the strictly
binary all of whose leaves are at level d.
A
B
N
C
G
O
1
0
1
2
3 3L
F
M
2
3 3H
D
I
2
3 J
E
K
2
3

A
B
Level 0: 20 nodes
H
D
I
E
J K
C
L
F
M
G
N O
Level 1: 21 nodes
Level 2: 22 nodes
Level 3: 23 nodes

• At level k, there are 2k nodes.
• Total number of nodes in the tree of depth
d:
20+ 21+ 22 + ………. + 2d =  2j = 2d+1 – 1
• In a complete binary tree, there are 2d leaf
nodes and (2d - 1) non-leaf (inner) nodes.
j=0
d

Operations on Binary Tree
• There are a number of operations that can
be defined for a binary tree.
• If p is pointing to a node in an existing tree
then
 left(p) returns pointer to the left subtree
 right(p) returns pointer to right subtree
 parent(p) returns the father of p
 brother(p) returns brother of p.
 info(p) returns content of the node.

Operations on Binary Tree
• In order to construct a binary tree, the
following can be useful:
• setLeft(p,x) creates the left child node of p.
The child node contains the info ‘x’.
• setRight(p,x) creates the right child node
of p. The child node contains the info ‘x’.

Applications of Binary Trees
• A binary tree is a useful data structure
when two-way decisions must be made at
each point in a process.
• For example, suppose we wanted to find
all duplicates in a list of numbers:
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

Applications of Binary Trees
• One way of finding duplicates is to
compare each number with all those that
precede it.
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

Searching for Duplicates
• If the list of numbers is large and is
growing, this procedure involves a large
number of comparisons.
• A linked list could handle the growth but
the comparisons would still be large.
• The number of comparisons can be
drastically reduced by using a binary tree.
• The tree grows dynamically like the linked
list.

• The binary tree is built in a special way.
• The first number in the list is placed in a
node that is designated as the root of a
binary tree.
• Initially, both left and right subtrees of the
root are empty.
• We take the next number and compare it
with the number placed in the root.
• If it is the same then we have a duplicate.

• Otherwise, we create a new tree node and
put the new number in it.
• The new node is made the left child of the
root node if the second number is less
than the one in the root.
• The new node is made the right child if the
number is greater than the one in the root.

14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14

15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
1415

15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14
15

4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14
15
4

4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14
154

9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14
154
9

7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14
154
9
7

18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14
154
9
7
18

3, 5, 16, 4, 20, 17, 9, 14, 5
14
154
9
7
18
3

3, 5, 16, 4, 20, 17, 9, 14, 5
14
154
9
7
183

5, 16, 4, 20, 17, 9, 14, 5
14
154
9
7
183
5

16, 4, 20, 17, 9, 14, 5
14
154
9
7
183
5
16

4, 20, 17, 9, 14, 5
14
154
9
7
183
5
16
4

20, 17, 9, 14, 5
14
154
9
7
183
5
16
20

20, 17, 9, 14, 5
14
154
9
7
183
5
16 20

17, 9, 14, 5
14
154
9
7
183
5
16 20
17

9, 14, 5
14
154
9
7
183
5
16 20
17

Binary Search Tree
• A binary tree with the property that items in
the left subtree are smaller than the root
and items are larger or equal in the right
subtree is called a binary search tree
(BST).
• The tree we built for searching for
duplicate numbers was a binary search
tree.
• BST and its variations play an important
role in searching algorithms.

Traversing a Binary Tree
• Suppose we have a binary tree, ordered
(BST) or unordered.
• We want to print all the values stored in
the nodes of the tree.
• In what order should we print them?

• Ways to print a 3 node tree:
14
154
(4, 14, 15), (4,15,14)
(14,4,15), (14,15,4)
(15,4,14), (15,14,4)

• In case of the general binary tree:
node
(L,N,R), (L,R,N)
(N,L,R), (N,R,L)
(R,L,N), (R,N,L)
left
subtree
right
subtr
ee
L
N
R

• Three common ways
node
Preorder: (N,L,R)
Inorder: (L,N,R)
Postorder: (L,R,N)
left
subtree
right
subtre
e
L
N
R

void preorder(TreeNode<int>* treeNode)
{
if( treeNode != NULL )
{
cout << *(treeNode->getInfo())<<" ";
preorder(treeNode->getLeft());
preorder(treeNode->getRight());
}
}

void inorder(TreeNode<int>* treeNode)
{
{
inorder(treeNode->getLeft());
inorder(treeNode->getRight());
}
}

void postorder(TreeNode<int>* treeNode)
{
{
postorder(treeNode->getLeft());
postorder(treeNode->getRight());
}
}

cout << "inorder: "; preorder( root);
cout << "inorder: "; inorder( root );
cout << "postorder: "; postorder( root );

Preorder: 14 4 3 9 7 5 15 18 16 17 20
14
154
9
7
183
5
16 20
17

Inorder: 3 4 5 7 9 14 15 16 17 18 20
14
154
9
7
183
5
16 20
17

Deleting a node in BST
• As is common with many data structures,
the hardest operation is deletion.
• Once we have found the node to be
deleted, we need to consider several
possibilities.
• If the node is a leaf, it can be deleted
immediately.

• If the node has one child, the node can be
deleted after its parent adjusts a pointer to
bypass the node and connect to inorder
successor.
6
2
4
3
1
8

• The inorder traversal order has to be
maintained after the delete.
6
2
4
3
1
8
6
2
4
3
1
8


• The inorder traversal order has to be
maintained after the delete.
6
2
4
3
1
8
6
2
4
3
1
8
6
2
31
8
 

• The complicated case is when the node to
be deleted has both left and right subtrees.
• The strategy is to replace the data of this
node with the smallest data of the right
subtree and recursively delete that node.

Delete(2): locate inorder successor
6
2
5
3
1
8
4
Inorder
successor

Delete(2): locate inorder successor
6
2
5
3
1
8
4
Inorder
successor
 Inorder successor will be the left-most
node in the right subtree of 2.
 The inorder successor will not have a left
child because if it did, that child would be
the left-most node.

Delete(2): copy data from inorder successor
6
2
5
3
1
8
4
 6
3
5
3
1
8
4

Delete(2): remove the inorder successor
6
2
5
3
1
8
4
 6
3
5
3
1
8
4
 6
3
5
3
1
8
4

Delete(2)
 6
3
5
4
1
8
 6
3
5
3
1
8
4

Week 8 (trees)

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Week 8 (trees)