presentation of data structure

8/7/2019 presentation of data structure

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Bismillah


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Presentation

Muhammad Irfan Khan (S-31021)Sheraz Khan (S-31030)

BE(SE) -16

Data structure


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represented via a tree data

structure

leaf nodes

root nodepresident

VP VP VP VP

mgr mgr

mgr

mgr


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Binary Trees

A binary tree is defined recursively. Eitherit is empty (the base case) or it consists of

a rootand two subtrees, each of which is a

binary tree

Binary trees and trees typically are

represented using references as dynamic

links, though it is possible to use fixedrepresentations like arrays

leaf nodes

root node


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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


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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 thatrequire searching, linear data structures are not

suitable.

We will focus our attention on binary trees.


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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 thetree.


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Binary Tree

Binary tree with 9 nodes.

A

B

D

H

C

E F

G I


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Binary Tree

A

B

D

H

C

E F

G I

Left subtree

root

Right subtree


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Binary Tree

Recursive definition

A

B

D

H

C

E F

G ILeft subtree

root

Right subtree


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Binary Tree


A

B

D

H

C

E F

G Iroot


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Binary Tree


A

B

D

H

C

E F

G I

root

Right subtree


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Binary Tree


A

B

D

H

C

E F

G I

root

Right subtreeLeft subtree


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Not a Tree

Structures that are not trees.

A

B

D

H

C

E F

G I


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Not a Tree


A

B

D

H

C

E F

G I


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Not a Tree


A

B

D

H

C

E F

G I


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Binary Tree: Terminology

A

B

D

H

C

E F

G I

parent

Left descendant Right descendant

Leaf nodes Leaf nodes


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Binary Tree

If every non-leaf node in a binary tree has non-emptyleft and right subtrees, the tree is termed a strictlybinary tree.

A

B

D

H

C

E F

G I

J

K


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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.


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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


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Complete Binary Tree

A complete binary tree of depth d is the strictly binaryall 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


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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


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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


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If the tree is built out of n nodes then

n = 2d+1 1

or log2(n+1) = d+1or d = log2(n+1) 1

I.e., the depth of the complete binary tree built using nnodes will be log2(n+1) 1.

For example, for n=100,000, log2(100001) is less than20; the tree would be 20 levels deep.

The significance of this shallowness will becomeevident later.


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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.


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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.


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In order to construct a binary tree, the following

can be useful:

setLeft(p,x) creates the left child node of p. Thechild node contains the info x.

setRight(p,x) creates the right child node of p.

The child node contains the info x.


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Applications ofBinary 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


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Applications ofBinary 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


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Searching for Duplicates

If the list of numbers is large and is growing,this procedure involves a large number ofcomparisons.

A linked list could handle the growth but thecomparisons 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.


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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.


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Otherwise, we create a new tree node and put

the new number in it.

The new node is made the left child of the rootnode 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.


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

14


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

1415


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

14

15


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

14

15

4


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

14

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

14

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9


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

14

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9


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

14

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7


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

14

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

14

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18


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3, 5, 16, 4, 20, 17, 9, 14, 5

14

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5, 16, 4, 20, 17, 9, 14, 5

14

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5


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5, 16, 4, 20, 17, 9, 14, 5

14

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5


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16, 4, 20, 17, 9, 14, 5

14

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5

16


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16, 4, 20, 17, 9, 14, 5

14

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5

16


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4, 20, 17, 9, 14, 5

14

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20, 17, 9, 14, 5

14

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20, 17, 9, 14, 5

14

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16 20


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17, 9, 14, 5

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16 20

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17, 9, 14, 5

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5

16 20

17


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9, 14, 5

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5

16 20

17


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Trace of insert

17, 9, 14, 5

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5

16 20

17pq


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Trace of insert

17, 9, 14, 5

14

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5

16 20

17p

q


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Trace of insert

17, 9, 14, 5

14

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5

16 20

17

p

q


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Trace of insert

17, 9, 14, 5

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16 20

17

p

q


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Trace of insert

17, 9, 14, 5

14

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16 20

17

pq


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Trace of insert

17, 9, 14, 5

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16 20

17

p

q


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Trace of insert

17, 9, 14, 5

14

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5

16 20

17

pq


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Trace of insert

17, 9, 14, 5

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16 20

17

p

q


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Trace of insert

17, 9, 14, 5

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16 20

17

p

p->setRight( node );

node


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Trace of insert

17, 9, 14, 5

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5

16 20

17

p

p->setRight( node );

node


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Binary Search Tree

A binary tree with the property that items in theleft subtree are smaller than the root and itemsare 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 insearching algorithms.


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Traversing a Binary Tree

Ways to print a 3 node tree:

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(4, 14, 15), (4,15,14)

(14,4,15), (14,15,4)

(15,4,14), (15,14,4)


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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)

leftsubtree

rightsubtree

L

N

R


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Three common ways

node

Preorder: (N,L,R)

Inorder: (L,N,R)

Postorder: (L,R,N)

leftsubtree

rightsubtreeL

N

R


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

14

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5

16 20

17


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

14

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5

16 20

17


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Postorder: 3 5 7 9 4 17 16 20 18 15 14

14

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5

16 20

17


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Level-order Traversal

There is yet another way of traversing a binary

tree that is not related to recursive traversal

procedures discussed previously.

In level-order traversal, we visit the nodes at

each level before proceeding to the next level.

At each level, we visit the nodes in a left-to-right

order.


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There is yet another way of traversing a binary

tree that is not related to recursive traversal

procedures discussed previously.

In level-order traversal, we visit the nodes at

each level before proceeding to the next level.

At each level, we visit the nodes in a left-to-right

order.


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How do we do level-order traversal?

Surprisingly, if we use a queue instead of a

stack, we can visit the nodes in level-order.

Here is the code for level-order traversal:


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Queue: 14

Output:

14

4

9

7

3

5

15

18

16 20

17


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Queue: 4 15

Output: 14

14

4

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7

3

5

15

18

16 20

17


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Queue: 15 3 9

Output: 14 4

14

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3

5

15

18

16 20

17


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Queue: 3 9 18

Output: 14 4 15

14

4

9

7

3

5

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18

16 20

17


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Queue: 9 18

Output: 14 4 15 3

14

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3

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15

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16 20

17


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Queue: 7 16 20

Output: 14 4 15 3 9 18

14

4

9

7

3

5

15

18

16 20

17


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Queue: 16 20 5

Output: 14 4 15 3 9 18 7

14

4

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3

5

15

18

16 20

17


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Queue: 20 5 17

Output: 14 4 15 3 9 18 7 16

14

4

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3

5

15

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16 20

17


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Queue: 5 17

Output: 14 4 15 3 9 18 7 16 20

14

4

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3

5

15

18

16 20

17


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Queue:

Output: 14 4 15 3 9 18 7 16 20 5 17

14

4

9

7

3

5

15

18

16 20

17


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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.


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If the node has one child, the node can be deleted afterits parent adjusts a pointer to bypass the node andconnect to inorder successor.

6

2

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3

1

8


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The inorder traversal order has to be maintained

after the delete.

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after the delete.

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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.


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Delete(2): locate inorder successor

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4Inorder

successor

S


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6

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4Inorder

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.

D l i d i BST


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Delete(2): copy data from inorder successor

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D l ti d i BST


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Delete(2): remove the inorder successor

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D l ti d i BST


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Delete(2)

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D l ti d i BST


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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.

D l ti d i BST


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If the node has one child, the node can be deleted afterits parent adjusts a pointer to bypass the node andconnect to inorder successor.

6

2

4

3

1

8

D l ti d i BST


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after the delete.

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D l ti d i BST


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after the delete.

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6

2

31

8

D l ti d i BST


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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.

D l ti d i BST


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6

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4Inorder

successor

D l ti d i BST


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6

2

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3

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4Inorder

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.

D l ti d i BST


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Delete(2): copy data from inorder successor

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1

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6

3

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1

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4



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Delete(2): remove the inorder successor

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presentation of data structure

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