Foundation of Computing Systems

Lecture 6

Trees: Part III

05.08.09 IT 60101: Lecture #6 2

Heap Trees

• A heap tree, say H, is a complete binary tree. H will be termed as heap tree, if it satisfies the following properties:

• (i) For each node N in H, the value at N is greater than or equal to the value of each of the children of N.

• (ii) Or in other words, N has the value which is greater than or equal to the value of every successor of N.

• Such a heap tree is called max heap.

• Similarly, min heap is possible, where any node N has the value less than or equal to the value of any of the successors of N.

05.08.09 IT 60101: Lecture #6 3

Heap Trees: Example

95

85 45

75

55 65

2535 15

(a ) M ax heap

15

45 25

55

85 95

6535 75

(b ) M in heap

05.08.09 IT 60101: Lecture #6 4

Array Representation of Heap Trees

1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 45 25 55 65 35 75 85 95 . . . . . .

15

45 25

55

85 95

6535 75

(b ) M in heap

05.08.09 IT 60101: Lecture #6 5

Operations on Heap Trees

• Operations important to heap is

• Insertion of a node

• Deletion of a node

• Merging two heaps

• Other operations are same as in general binary trees

05.08.09 IT 60101: Lecture #6 6

Heap Trees: Insertion

95

85 45

75 25 35 15

55 65 19

(b ) Inc lus ion o f 19 in the fash ion o f com p le teb ina ry tree and it sa tis fy the p roperty o f heap

95

85 45

75

55 65

25 35 15

(a ) M ax heap

A trivial case

05.08.09 IT 60101: Lecture #6 7

Heap Trees: Insertion

A non-trivial case

95

85 45

75 25 15

65 11155 19

111

111 95

11185

25

35

111

95 45

75 85 35 15

55 65 19 25

W hen 111 is inse rted in to the heap tree

05.08.09 IT 60101: Lecture #6 8

Heap Trees: Deletion

• Any node can be deleted from a heap tree. But from the application point of view, deleting the root node has some special importance.

• Copy the root node into a temporary storage, say ITEM.

• Replace the root node by the last node in the heap tree. Then reheap the tree as stated below:

– Let newly modified root node be the current node. Compare its value with the value of its two child. Let X be the child whose value is the largest. Interchange the value of X with the value of the current node.

– Make X as the current node.

• Continue reheap, if the current node is not an empty node.

05.08.09 IT 60101: Lecture #6 9

Heap Trees: Deletion

45 63

27 12

293542

24

26

57

H eap tree after repalc ing 99 by 26

99

45 63

27 12

293542

24

26

57

26

D ele ting the node w ith data 99

05.08.09 IT 60101: Lecture #6 10

Heap Trees: Deletion

45

63

27 12

293542

24

26

57

H eap tree a fter rebu ild

45 63

27 12

293542

24

63

26

57

R ebuild ing a fter ad justing 63

05.08.09 IT 60101: Lecture #6 11

Heap Trees: Deletion

45

63

27 12

293542

24

26

57

H eap tree a fter rebu ild

45

27 12

293542

24

63

2657

5726

R ebuild the tree a t 26

05.08.09 IT 60101: Lecture #6 12

Heap Trees: Deletion

45

63

57

26 4235 29

27 12 24

A fter de le tion o f 99

99

45 63

27 12

293542

24

26 63

2657

57

26

26

D ele ting the node w ith data 99

05.08.09 IT 60101: Lecture #6 13

Heap Trees: Merging

92

45 67

5938

19

13

80

9692 93

96

93

67

80 9213 19

67

4538 59

H : A m ax heap H : A m in heap

R esultant m ax heap after m erg ing H and H

+1 2

1 2

05.08.09 IT 60101: Lecture #6 14

Application of Heap Trees

• Sorting

• Priority queue

05.08.09 IT 60101: Lecture #6 15

Application of Heap Trees: Sorting

• Step 1: Build a heap tree with the given set of data.

• Step 2:• (a) Delete the root node from the heap.

• (b) Rebuild the heap after the deletion.

• (c) Place the deleted node in the output.

• Step 3: Continue Step 2 until the heap tree is empty.

05.08.09 IT 60101: Lecture #6 16

Sorting with Heap Tree: Example

33, 14, 65, 02, 76, 69, 59, 85, 47, 99, 98

99

98 96

65 33 59

47 76

85

02 14

99 98 69 65 85 33 59 02 47 14 76

Building (max) heap tree from the given set of data

05.08.09 IT 60101: Lecture #6 17

Sorting with Heap Tree: Example

99

98 96

65 33 59

47 76

85

02 14

99 98 69 65 85 33 59 02 47 14 76

76

98 69

65 85 33 59

02 47 14 99

9976 98 69 65 85 33 59 02 47 14

Swapping the root and the last node

05.08.09 IT 60101: Lecture #6 18

Sorting with Heap Tree: Example

76

98 69

65 85 33 59

02 47 14 99

9976 98 69 65 85 33 59 02 47 14

98

85 69

65 76 33 59

02 47 14

9998 85 69 65 76 33 59 02 47 14

Rebuild the heap tree

05.08.09 IT 60101: Lecture #6 19

Sorting with Heap Tree: Example

98

85 69

65 76 33 59

02 47 14

9998 85 69 65 76 33 59 02 47 14

85

76 69

65 14 33 59

02 47

9985 76 69 65 14 33 59 02 47 98

Continuation of Step 2 after selecting 98

05.08.09 IT 60101: Lecture #6 20

Sorting with Heap Tree: Example

02 14 33 47 59 65 69 76 85 98 99

Sorted list when the heap is empty

05.08.09 IT 60101: Lecture #6 21

Priority Queue with Heap Tree

Consider the following processes with their priorities:

Process P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Priority 5 4 3 4 5 5 3 2 1 5

Order of arrivals of processes are:

P1    P5    P6    P10    P2    P4    P3    P7    P8    P9

05.08.09 IT 60101: Lecture #6 22

Priority Queue with Heap Tree

P(5)

P(5) P (5)

P(4) P (5) P(3) P(3)

P(2) P(1) P(4)

1

2

34

5 6

7

8 9

10

(a ) P rio rity queue heap (subsc rip t ind ica tes o rde ro f p rocess w he reas num ber in pa ran theses

m eans its p rio rity)

P(5)

P(5) P (5)

P(4) P (4) P(3) P(3)

P(2) P(1)

1

2 34

5

6

7

8 9

10

(b ) A fte r the rem ova l o f P

05.08.09 IT 60101: Lecture #6 23

Threaded Binary Trees

• In a linked representation of binary trees with n (n > 0) nodes, n + 1 link fields contain null entries.

+

- *

A B C /

D E

05.08.09 IT 60101: Lecture #6 24

Threaded Binary Trees

• Issues:• How a threaded binary tree will be represented

and how can we distinguish links from threads.

• What advantages can be obtained from threaded binary trees.

• How the different operations like insertion, deletions, traversals, etc., can be carried out on it.

05.08.09 IT 60101: Lecture #6 25

Threaded Binary Trees

• Issues:• How a threaded binary tree will be represented

and how can we distinguish links from threads.

– Threading according to a tree traversal» Inorder Inorder threading» Preorder Preorder threading» Postorder Postorder threading

– Another a bit field is required to distinguish a thread and link

05.08.09 IT 60101: Lecture #6 26

Threaded Binary Trees: Example

A

B C

D

E F

G

Inorder T raversa l : B A G E D F C

Inorder threading of a binary tree

05.08.09 IT 60101: Lecture #6 27

Threaded Binary Trees: Example

A

B C

D

E F

G

LCHILD LTAG DATA RTAG RCHILD

05.08.09 IT 60101: Lecture #6 28

Advantages of a Threaded Binary Tree

• The traversal operation is more faster than that of its unthreaded version.

• Efficiently determine the predecessor and successor nodes starting from any node.

• Any node can be accessible from any other node.

• Insertions into and deletions from a threaded tree are although time consuming operations (since we have to manipulate both links and threads) but these are very easy to implement.

05.08.09 IT 60101: Lecture #6 29

Operations on Threaded Binary Tree

For various operations on threaded binary trees see the book

Classic Data StructuresChapter 7

PHI, 2nd Edn., 17th Reprint