Chapter VI Tree - SI-35-02 · A connected graph that contains no simple circuits is called a tree Trees were used since 1857, when English mathematician Arthur Cayley used them to

Tree

Discrete Mathematics (MA 2333)Faculty of Science Telkom Institute of TechnologyBandung - Indonesia

Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology

DefinitionA connected graph that contains no simple circuits is called a treeTrees were used since 1857, when English mathematician Arthur Cayley used them to count certain types of chemical compoundsApplications in computer science :

Constructing efficient algorithms for locating items in the listTree can be used in Huffman coding, that construct efficient codes saving costs in data transmission and storageCompilation techniques, and so on


Definition

a b

c d

e f

a b

c d

e f

a b

c d

e f

a b

c d

e f

Tree Tree Not Tree Not Tree


DefinitionForest is disconnected graph that each of their connected components is a tree

Forest that contains thee trees


Properties of Trees

Given Tree with n vertexThere is a unique simple path between any two of its verticesThe tree has n-1 edges


Spanning TreeSpanning tree is the spanning subgraf in the form of tree Spanning tree is obtained by eliminating the circuit on the graph

G T1 T2 T3 T4

Every connected graph has at least a spanning tree


Applications of Spanning Tree

Constructing topology of a computer network that all host still connected, with minimum cost

Example

(a) (b)Router

Subnetwork


Minimum Spanning Tree

Connected-weighted graph has at least a spanning treeSpanning tree having minimum weight is called minimum spanning tree

a

bc

d

e

f

g

h55

5

40

25

45

30

5020

15

35 10

a

bc

d

e

f

g

h

5

40

25 30

20

15

10


Prim’s Algorithm

1. Choose a minimum weighted edge of the graph G. Insert into T

2. Choose edge (u, v) on G which has a minimum weight and is incident to vertex on T. This edge does not form a circuit on T. Insert (u, v) into T

3. Repeat step 2 n-2 times


Prim’s AlgorithmDetermine minimum spanning tree of the graph presented below

A B

C

D

E

2

1,2

0,91,3

1,6

0,80,7

2,2

1

1,4


Prim’s Algorithm

0,8(C,E)2

0,7(B,E)1

Spanning TreeWeightEdgeStep

B

E

B

E

C


Prim’s Algorithm

0,9(A,D)4

1,2(B,A)3


B

E

CA

B

E

CA

D


Prim’s Algorithm

Minimum Spanning Tree resulted:

with total weight : 0,7 + 0,8 + 1,2 + 0,9 = 3,6


Kruskal’s Algorithm

1. Step 0 : Sort (ascending) all edges of the graph by weight

2. Step 1: T still empty 3. Step 2: choose (u, v) which has

minimum weight and does not form a circuit into T

4. Step 3: repeat step 2 n – 1 times


Kruskal’s AlgorithmDetermine minimum spanning tree of the graph presented below

A B

C

D

E

2

1,2

0,91,3

1,6

0,80,7

2,2

1

1,4



Edges (B,E) (C,E) (A,D) (B,C) (A,B) (B,D) (D,E) (C,D) (A,C) (A,E) Weight 0,7 0,8 0,9 1 1,2 1,3 1,4 1,6 2 2,2

0,7(B,E)1Spanning TreeWeightEdgeStep

B

E



0,9(A,D)3

0,8(C,E)2


B

E

C

B

E

CA

D



1,2(A,B)5

Rejected1(B,C)4


B

E

CA

D



Minimum Spanning Tree resulted:

with total weight : 0,7 + 0,8 + 1,2 + 0,9 = 3,6


Rooted Tree

A rooted tree is a tree in which one vertex has been designated as the root and every adge is directed away from the root

Rooted Tree Removing directions


Pohon Berakar

a

b

c

d

e f

g

h

f

g

a

b

cd

e

f

g h

d

e

hb

a c

b as root e as root


Terminology on Rooted Tree1. Child or children

and parentb, c, and d are child of a, a is parent of b, c, d

2. PathPath from a to i is a, b, e, i. length of path from a to j is 3.

a

bc

d

ef g

h i j


Terminology on Rooted Tree3. Sibling

i sibling of j, g is not sibling of e, since they have different parent

4. Subtree

a

bc

d

ef g

h i j


Terminology on Rooted Tree4. Degree

Degree of the vertex is the number of children on it degree of a is 3, degree of d is 1 degree of j is 0. Maximum degree of all vertices is the degree of the tree itselfThis tree has degree= 3


Terminology on Rooted Tree5. Leaf

Vertex with degree 0 (or having no children) is called leafVertex h, i, j, f, c, g are leaves

6. Internal vertexVertex having children is called internal nodesVertex b, c, d, e are internal vertex


Terminology on Rooted Tree7. Level 8. Height or depth

Maximum level of a tree is called height or dept of the tree. This tree has height 3

a

b

cd

ef g

h i j

0

1

2

3


m-ary Tree

Rooted tree which its internal vertex having maximum n children is called n-ary tree. If n=2, its tree is called binary tree m-ary tree is called regular or full if for each internal vertex have exactly m child


Binary Tree

a

b c

d

a

b c

d


Binary Tree

a

b

c

d

a

b

c

d

Right-skewed Tree Full Binary TreeLeft-skewed Tree


Tree Traversal

Rooted tree are often used to store information. We need procedures for visiting each vertex of an ordered rooted tree to access data. We will describe several important algorithms for visiting all the vertices of an rooted tree


Tree Traversal

PreorderIn OrderPost Order


Preorder1. Visit root2. Visit left subtree3. Visit right subtree

Given Binary Tree


Preorder

a b c

d

g h

e f


Preorder

a b c d e f

g h


Inorder1. Visit left subtree2. Visit root3. Visit right subtree

Given Binary Tree


Inorder

ab c

d

g h

e f


Inorder


Postorder1. Visit left subtree2. Visit right subtree3. Visit root

Given Binary Tree


Postorder

ab ed

g h

cf


Postorder


Applications of Binary Tree

Expression TreeBinary Search TreeHuffman Code


Expression Tree

Expression Tree of (a/(b*(c/d)))*((e+f)*g)

a

b

d

e

c

*

f g


Expression Tree

Expression tree is used by high level language compiler to evaluate expression which written in infix notation, prefix notation, and postfix notationInfix notation

form : <operand1> <operator> <operand2>

example : a + bfully parenthesized expressionobtained by traverse expression tree in inorder


Expression Tree

Prefix notationform : <operator> <operand1> <operand2>

example : + a bobtained by traverse expression tree in preorder

Postfix notationform : <operand1> <operand2> <operator>

example : a b +obtained by traverse expression tree in postorder


Expression Tree

Infix notation :(a/(b*(c/d)))*((e+f)*g)Prefix notation :*/a*b/cd*+efgPostfix notation :abcd/*/ef+g**


Huffman Coding

Huffman coding is fundamental algorithm in data compression, the subject devoted to reducing the number of bits required to represent informationHuffman coding is extensively used to compress bit strings representing text and it also plays an important role in compressing audio and image files


Huffman Coding

Algorithm that takes as input the frequencies (which are probability of occurrences) of symbols in a string and produces as output a prefix codes for these symbols


Huffman CodingFixed-length code (take from Rinaldi Munir)

Characters a b c d e f----------------------------------------------------------------

Frequencies 45% 13% 12% 16% 9% 5%Code 000 001 010 011 100 111

‘bad’ coded by ‘001000011’

Coding 100.000 characters needs 300.000 bits.


Huffman CodingVariable-length code (Huffman code)

Characters a b c d e f------------------------------------------------------------------------

Frequencies 45% 13% 12% 16% 9% 5%Code 0 101 100 111 1101 1100

‘bad’ coded by ‘1010111 ’

coding 100.000 characters needs (0,45 × 1 + 0,13 × 3 + 0,12 × 3 + 0,16 × 3 +0,09 × 4 + 0,05 × 4) × 100.000 = 224.000 bit

Compression ratio:

(300.000 – 224.000)/300.000 × 100% = 25,3%


Huffman Coding


c:12 b:13

cb:25

f:5 e:9

fe:14 d:16

fed:30 a:454.

Huffman Coding


cbfed:55

c:12 b:13

cb:25

f:5 e:9

fe:14 d:16

fed:30

a:455

Huffman Coding


Huffman Coding

cbfed:55

c:12 b:13

cb:25

f:5 e:9

fe:14 d:16

fed:30

a:45

acbfed:1000 1

0 1

0 1 0 1

0 1

6


Binary Search Trees (BST)

Searching for items is a list is one of the most important task that arises in computer science.Our primary goal is to implement a searching algorithm that finds item efficiently when the items are totally ordered. This can be accomplished through binary search tree



Each child of a vertex designated as right or left childNo vertex has more than one right child or left childEach vertex labeled with a key, which is one of the itemsVertices are assigned keys so that the key of a vertex is both larger than the keys of all vertices in its left subtree and smaller than keys of all vertices in its right subtree



Given data which ordered by input:

65, 34, 22, 45, 100, 120, 8

The BST is


Exercises1. Determine minimum spanning tree of graph below. Use

Prim’s algorithm, and Kruskal’s algorithm!


Exercises

2. Given prefix notation : /+*AB/CD*E-FG

Construct a expression tree by this notationWhat is infix notation?What is post notation?


Exercises

3. Given data which ordered by input :45, 78, 3, 100, 60, 80, 120

Construct BST!Traverse the BST in preorderTraverse the BST in inorderTraverse the BST in postorder


References

Munir, Rinaldi, Matematika DiskritEdisi 2, Penerbit InformatikaBandung, Bandung, 2003 Rosen, Kenneth H., Discrete Mathematics and Its Applications 5th Ed, McGraw-Hill, New York, 2003

Documents

Chapter VI Tree - SI-35-02 · A connected graph that contains no simple circuits is called a tree Trees were used since 1857, when English mathematician Arthur Cayley used them to