discrete math- tree

7/27/2019 Discrete Math- Tree

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IEEM 2021 Discrete Mathematics 1

Chapter 9

Trees



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9.1: Introduction to Trees9.1: Introduction to TreesDefinition 1Definition 1 (P.631): A(P.631): A treetree is ais a connectedconnected undirectedundirected

graph that contains no circuits.graph that contains no circuits.

Theorem 1 (P632):Theorem 1 (P632): There is a unique simple path between anyThere is a unique simple path between any

two of its nodes.two of its nodes.

A (notA (not--necessarilynecessarily--connected) undirected graph withoutconnected) undirected graph without

simple circuits is called asimple circuits is called a forestforest..You can think of it as a set of trees having disjoint sets ofYou can think of it as a set of trees having disjoint sets of

nodes.nodes.

AA leafleafnode in a tree or forest is any pendant or isolatednode in a tree or forest is any pendant or isolated

vertex. Anvertex. An internalinternalnode is any nonnode is any non--leaf vertex (thus itleaf vertex (thus it

has degreehas degree ___ ).___ ).


Tree and Forest ExamplesTree and Forest Examples

A Tree:A Tree: A Forest:A Forest:

Leaves in green, internal nodes in brown.


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family tree


ExampleExample 1 (P. 632): Which graphs are trees?1 (P. 632): Which graphs are trees?

not connected


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Rooted TreesRooted Trees

Definition 2Definition 2 (P.633): A(P.633): A rooted treerooted tree is a tree inis a tree in

which one node has been designated thewhich one node has been designated the rootroot.. Every edge is (implicitly or explicitly) directed awayEvery edge is (implicitly or explicitly) directed away

from the root.from the root.

You should know the following terms aboutYou should know the following terms about

rooted trees:rooted trees:

Parent, child, siblings, ancestors, descendents, leaf,Parent, child, siblings, ancestors, descendents, leaf,

internal node,internal node, subtreesubtree..



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Rooted Tree ExamplesRooted Tree Examples

Note that a givenNote that a given unrootedunrooted tree withtree with nn nodesnodes

yieldsyields nn different rooted trees.different rooted trees.

root

root

Same tree

except

for choice

of root


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RootedRooted--Tree Terminology ExerciseTree Terminology Exercise

Find theFind theparentparent,,childrenchildren,,siblingssiblings,,

ancestorsancestors, &, &

descendantsdescendants

of nodeof nodeff.. a

b

c

d

e

f

g

h

i

j k l

m

no

p

q

r

root


Example 2Example 2 (P. 634): Find(P. 634): Find

the parent ofthe parent ofcc..

the children ofthe children ofgg..

the siblings ofthe siblings ofhh..

all ancestors ofall ancestors ofee..all descendants ofall descendants ofbb..

all internal vertices.all internal vertices.

all leavesall leaves

subtreesubtree rooted atrooted at atatgg


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Parent ofParent ofgg:: aa

Child ofChild ofgg:: hh,, ii,,jj

Siblings:Siblings: hh,, ii,,jj are sibling.are sibling.

Ancestors ofAncestors ofjj:: aa,,gg((aa has no ancestor.)has no ancestor.)

Descendents ofDescendents ofgg:: hh,, ii,,jj,, kk,, ll,, m.m.

Leaf:Leaf: dd,,ee,,ff,, kk,, ii,, ll,, mm..

Internal node:Internal node: VV{leaf vertices}{leaf vertices}

SubtreeSubtree with rootwith rootgg::


nn--aryary treestrees

Definition 3Definition 3 (P.634): A rooted tree is called(P.634): A rooted tree is called

nn--aryary if every vertex has no more thanif every vertex has no more than nn

children.children.

It is calledIt is calledfullfullif every internal (nonif every internal (non--leaf) vertexleaf) vertexhashas exactlyexactly nn children.children.

A 2A 2--ary tree is called aary tree is called a binary treebinary tree..

These are handy for describing sequences of yesThese are handy for describing sequences of yes--

no decisions.no decisions.

Example:Example: Comparisons in binary search algorithm.Comparisons in binary search algorithm.


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Which Tree is Binary?Which Tree is Binary?

A given rooted tree is a binary treeA given rooted tree is a binary tree iffiffevery nodeevery node

other than the root has degreeother than the root has degree

___, and the root has___, and the root hasdegreedegree ___.___.


Example 3Example 3 (P. 634): Are the following rooted(P. 634): Are the following rooted

trees fulltrees full mm--aryary trees for some positive integertrees for some positive integermm..

full binary

tree

full 3-ary

tree

full 5-ary

tree

no


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Ordered Rooted TreeOrdered Rooted Tree

This is just aThis is just a rooted treerooted tree in which thein which the

children of each internal node are ordered.children of each internal node are ordered.

In ordered binary trees, we can define:In ordered binary trees, we can define:

left child, right childleft child, right child

leftleft subtreesubtree, right, right subtreesubtree

ForFornn--aryary trees withtrees with nn > 2> 2, can use terms, can use terms

likelike leftmostleftmost,, rightmost,rightmost, etc.etc.


Example 4Example 4 (P. 635): Find(P. 635): Find

(1)(1) the left children ofthe left children ofdd..

(2)(2) the right children ofthe right children ofdd..

(3)(3) the leftthe left subtreesubtree ofofcc..

(4)(4) the rightthe right subtreesubtree ofofcc..SolutionSolution::

the left children ofthe left children ofdd::ff

the right children ofthe right children ofdd::gg

leftleft subtreesubtree ofofcc rightright subtreesubtree ofofcc


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Trees as ModelsTrees as Models

Can use trees to model the following:Can use trees to model the following:

Saturated hydrocarbonsSaturated hydrocarbons

Organizational structuresOrganizational structures

Computer file systemsComputer file systems

In each case, would you use a rooted or aIn each case, would you use a rooted or a

nonnon--rooted tree?rooted tree?


Example 5Example 5 (P.636):(P.636): Saturated hydrocarbonsSaturated hydrocarbons

Arthur Cayley

1821 - 1895

There are exactly two different isomers of C4H10.


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Example 6Example 6 (P.637):(P.637): Representing OrganizationRepresenting Organization


Example 6Example 6 (P.637):(P.637): Computer File SystemsComputer File Systems


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Example 7Example 7 (P.638):(P.638): TreeTree--Connected Parallel ProcessorsConnected Parallel Processors

1st step:

P4: x1 +x2

P5: x3 +x4P6: x5 +x6

P7: x7 +x8

2nd step:

P2: (x1 +x2) + (x3 +x4)

P3: (x5 +x6) + (x7 +x8)

3rd step:

P1: [(x1+x2)+(x3+x4)]+[(x5+x6)+(x7+x8)]


Example 7Example 7 (P.638):(P.638): TreeTree--Connected Parallel ProcessorsConnected Parallel Processors

1st step:

P4: x1 +x2

2nd step:

P3: (x5 +x6) + (x7 +x8)

3rd step:

P1: [(x1+x2)+(x3+x4)]+[(x5+x6)+(x7+x8)]

2nd step:

P2: (x

1+x

2) + (x

3+x

4)

1st step:

P7: x7 +x81st step:

P6

: x5

+x6

1st step:

P5

: x3

+x4


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Properties of TreesProperties of Trees

Theorem 2Theorem 2 (P. 638): Any tree with(P. 638): Any tree with nn nodes hasnodes has ee == nn 11edges.edges.

Proof:Proof: By Mathematical Induction.By Mathematical Induction.

Theorem 3Theorem 3 (P. 638):A full(P. 638):A full mm--aryary tree withtree with ii internalinternalnodes hasnodes has nn == mimi + 1+ 1 vertices, andvertices, and = (= (mm1)1)ii + 1+ 1 leaves.leaves.

Proof:Proof: There areThere are mimi children of internal nodes, plus the root.children of internal nodes, plus the root.

Thus,Thus, nn == mimi + 1+ 1. and,. and, == nn ii = (= (mm1)1)ii + 1+ 1..

Thus, whenThus, when mm is known and the tree is full, we canis known and the tree is full, we cancompute all four of the valuescompute all four of the values ee,, ii,, nn, and, and , given any, given any

one of them.one of them.


Theorem 4Theorem 4 (P. 639):(P. 639): A fullA full mm--aryary treetree

((ii)) nn verticesvertices

# of internal vertices# of internal vertices ii = (= (nn --1)/1)/m.m.

# of leaves# of leaves = [(= [(mm --1)1)nn + 1]/+ 1]/m.m.

((iiii)) ii internal verticesinternal vertices

# of vertices# of vertices nn == mimi +1.+1.

# of leaves# of leaves = (= (mm 1)1)ii + 1.+ 1.

((iiiiii)) leavesleaves

# of vertices# of vertices nn = (= (mm 1)/(1)/(mm 1).1).

# of internal vertices# of internal vertices ii = (= (ii --1)/(1)/(mm 1).1).


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Some More Tree TheoremsSome More Tree Theorems

Definition:Definition: TheThe levellevelof a vertex is the length of theof a vertex is the length of the

simple path from the root to the node.simple path from the root to the node.

TheThe heightheightof a tree is maximum of theof a tree is maximum of the

levels of vertices.levels of vertices.

A rootedA rooted mm--aryary tree with heighttree with height hh isis

calledcalled balancedbalancedif all leaves are atif all leaves are at

levelslevels hh ororhh11..

Example 10Example 10 (P.640): What is the level of(P.640): What is the level of

each node? What is the height of this tree?each node? What is the height of this tree?


Example 11Example 11 (P.640)(P.640):: Which trees are balanced?Which trees are balanced?

Theorem 5Theorem 5(P.640)(P.640):: There are at mostThere are at most mmhh leaves in anleaves in an

mm--aryary tree of heighttree of height hh..

ProofProof:: mathematical inductionmathematical induction


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Corollary 1Corollary 1 (P. 641)(P. 641):: AnAn mm--aryary tree withtree with

leaves has heightleaves has height hh loglogmm . If. Ifmm--aryary tree istree is

full and balanced thenfull and balanced then hh == loglogmm..

(i)(i)

(ii) If(ii) Ifmm--aryary tree is full,tree is full,

2

2

1 3 3

1 hm m m

+ +

+ + + +

# of leaves in a full m-ary tree

log .h ml m h l

log .h ml m h l = =


9.2: Applications of Trees9.2: Applications of Trees Binary search treesBinary search trees

A simple data structure for sorted listsA simple data structure for sorted lists

Decision treesDecision treesMinimum comparisons in sorting algorithmsMinimum comparisons in sorting algorithms

Prefix codesPrefix codes

Huffman codingHuffman coding

Game treesGame trees


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Binary Search TreesBinary Search Trees

A representation for sorted sets of items.A representation for sorted sets of items.

Supports the following operations inSupports the following operations in (log(log nn))averageaverage--case time:case time:

Searching for an existing item.Searching for an existing item.

Inserting a new item, if not already present.Inserting a new item, if not already present.

Supports printing out all items inSupports printing out all items in ((nn) time.) time.

Note that inserting into a plain sequenceNote that inserting into a plain sequence aaiiwould instead takewould instead take ((nn) worst) worst--case time.case time.


Binary Search Tree FormatBinary Search Tree Format

Items are stored atItems are stored atindividual tree nodes.individual tree nodes.

We arrange for theWe arrange for thetree to always obeytree to always obeythis invariant:this invariant:

For every itemFor every itemxx,,

Every node inEvery node inxxss leftleftsubtreesubtree is less thanis less thanxx..

Every node inEvery node inxxss rightrightsubtreesubtree is greater thanis greater thanxx..

7

3 12

1 5 9 15

0 2 8 11

Example:


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Example 1Example 1 (P. 644)(P. 644):: Form the binary search tree for theForm the binary search tree for the

words:words: mathematicsmathematics,,physicsphysics,,geographygeography,,zoologyzoology,,

meteorologymeteorology,,geologygeology,,psychologypsychology, and, and chemistrychemistry (using(using

alphabetical order).alphabetical order).

SolutionSolution::

not unique

May not be a balanced binary tree


Search on a binary tree:Search on a binary tree: If it present, we locate it.If it present, we locate it.

Otherwise, we try to add it to the binary search tree.Otherwise, we try to add it to the binary search tree.

In Algorithm 1, it locateIn Algorithm 1, it locatexx if it is already the key of aif it is already the key of a

vertex. Whenvertex. Whenxx is not a key, a new vertex with keyis not a key, a new vertex with keyxx isisadded to the tree. In theadded to the tree. In thepseudocodepseudocode,, vv is the vertexis the vertex

that hasthat hasxx as its key, andas its key, and label(label(vv) represents the key of) represents the key of

vertexvertex vv..


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Recursive Binary Tree InsertRecursive Binary Tree Insert


Suppose we have a binary search treeSuppose we have a binary search tree TTfor a list offor a list ofnn

items. We may form a full binary treeitems. We may form a full binary tree UUfromfrom TTbyby

adding unlabeled vertices whenever necessary so thatadding unlabeled vertices whenever necessary so that

every vertex with a key has two children.every vertex with a key has two children.


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Decision TreesDecision Trees

A decision tree represents aA decision tree represents a decisiondecision--makingmaking

processprocess.. Each possibleEach possible decision pointdecision point or situation isor situation is

represented by a node.represented by a node.

Each possible choice that could be made at thatEach possible choice that could be made at that

decision point is represented by an edge to a child node.decision point is represented by an edge to a child node.

In the extended decision trees used inIn the extended decision trees used in decisiondecision

analysisanalysis, we also include nodes that represent, we also include nodes that represent

random events and their outcomes.random events and their outcomes.


CoinCoin--Weighing ProblemWeighing Problem

Example 2Example 2 (P. 647): Suppose that(P. 647): Suppose that

you have 8 coins, one of which is ayou have 8 coins, one of which is a

lighter counterfeit, and a freelighter counterfeit, and a free--beambeam

balance.balance. (No scale of weight markings is(No scale of weight markings is

required for this problem!)required for this problem!)

How manyHow many weighingsweighings are needed toare needed to

guarantee that the counterfeit coinguarantee that the counterfeit coin

will be found?will be found?

?


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A DecisionA Decision--Tree ProblemTree Problem

In each situation, we pick two disjoint and equalIn each situation, we pick two disjoint and equal--

size subsets of coins to put on the scale.size subsets of coins to put on the scale.

The balance then

decides whether

to tip left, tip right,

or stay balanced.

A given sequence of

weighings thus yields

a decision tree with

branching factor 3.



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Applying the Tree Height TheoremApplying the Tree Height Theorem

The decision tree must have at least 8 leafThe decision tree must have at least 8 leaf

nodes, since there are 8 possible outcomes.nodes, since there are 8 possible outcomes.

In terms of which coin is the counterfeit one.In terms of which coin is the counterfeit one.

Recall the treeRecall the tree--height theorem,height theorem, hh loglogmm..

Thus the decision tree must have heightThus the decision tree must have height

hh loglog3388 == 1.8931.893 = 2= 2..

LetLet

s see if we solve the problem withs see if we solve the problem with

onlyonly

22 weighingsweighings


General Solution StrategyGeneral Solution Strategy

The problem is an example of searching for 1 unique particularThe problem is an example of searching for 1 unique particular

item, from among a list ofitem, from among a list ofnn otherwise identical items.otherwise identical items.

Somewhat analogous to the adage ofSomewhat analogous to the adage ofsearching for a needle in haystack.searching for a needle in haystack.

Armed with our balance, we can attack the problem using aArmed with our balance, we can attack the problem using adividedivide--andand--conquerconquer strategy, like whatstrategy, like whats done in binary search.s done in binary search.

We want to narrow down the set of possible locations where the dWe want to narrow down the set of possible locations where the desiredesired

item (coin) could be found down fromitem (coin) could be found down from nn to just 1, in a logarithmic fashion.to just 1, in a logarithmic fashion.

Each weighing has 3 possible outcomes.Each weighing has 3 possible outcomes.

Thus, we should use it to partition the search space into 3 piecThus, we should use it to partition the search space into 3 pieces that are ases that are as

close to equalclose to equal--sized as possible.sized as possible.

This strategy will lead to the minimum possible worstThis strategy will lead to the minimum possible worst--casecase

number ofnumber ofweighingsweighings required.required.


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General Balance StrategyGeneral Balance Strategy

On each step, putOn each step, put nn/3/3 of theof the nn coins to becoins to be

searched on each side of the scale.searched on each side of the scale.

If the scale tips to the left, then:If the scale tips to the left, then:

The lightweight fake is in the right set ofThe lightweight fake is in the right set ofnn/3/3 nn/3/3 coins.coins.

If the scale tips to the right, then:If the scale tips to the right, then:

The lightweight fake is in the left set ofThe lightweight fake is in the left set ofnn/3/3 nn/3/3 coins.coins.

If the scale stays balanced, then:If the scale stays balanced, then:

The fake is in the remaining set ofThe fake is in the remaining set ofnn 22nn/3/3 nn/3/3 coins thatcoins thatwere not weighed!were not weighed!

Except ifExcept ifnn mod 3 = 1mod 3 = 1 then we can do a little betterthen we can do a little betterby weighingby weighing nn/3/3 of the coins on each side.of the coins on each side.

You can prove that this strategy always leads to a balanced 3-ary tree.


Coin Balancing Decision TreeCoin Balancing Decision Tree

How many weighing steps?How many weighing steps? TT((nn) =) = TT((nn/3) +/3) + CC

Here is what the tree looks like in our case:Here is what the tree looks like in our case:

123 vs 456

1 vs. 2

left:123 balanced:

78right:

456

7 vs. 84 vs. 5

L:1 R:2 B:3 L:4R:5 B:6 L:7 R:8


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Example 3Example 3 (P.848): Give a decision tree that orders the(P.848): Give a decision tree that orders the

elements of the listelements of the list aa,, bb,, cc..

Theorem 1Theorem 1 (P.848): Any sorting algorithm based on(P.848): Any sorting algorithm based on

comparison requires at leastcomparison requires at least loglog22nn!! comparisons.comparisons.loglog22nn!! == ((nnloglog22nn))


Complexity of Sorting Algorithms Based on ComparisonsComplexity of Sorting Algorithms Based on Comparisons

Theorem 1Theorem 1 (P.848): Any sorting algorithm based on(P.848): Any sorting algorithm based on

comparison requires at leastcomparison requires at least log2log2nn!! comparisons.comparisons.

log2log2nn!! == ((nnloglog22nn))

Corollary 1Corollary 1 (P.848): The number of comparisons used by(P.848): The number of comparisons used by

a sorting algorithm to sort n elements based on binarya sorting algorithm to sort n elements based on binary

comparisons iscomparisons is ((nnloglog22nn).).

Theorem 2Theorem 2 (P.848): The average number of comparisons(P.848): The average number of comparisons

used by a sorting algorithm to sort n elements based onused by a sorting algorithm to sort n elements based on

binary comparisons isbinary comparisons is ((nnloglog22nn).).


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(Grundy's Game): n ,

, ,

, ,

, n = 7, (

), ,

,( 7 )

( 5, 1, 1) ( 4, 2, 1 )

( 5, 2 ) ( 4, 3 )( 6, 1 )

( 3, 2, 2 ) ( 3, 3, 1 )

( 4, 1, 1, 1 )

( 3, 2, 1, 1 ) ( 2, 2, 2, 1 )

( 3, 1, 1, 1, 1 )

( 2, 1, 1, 1, 1, 1 )

( 2, 2, 1, 1, 1 )

( 7 )

( 5, 1, 1) ( 4, 2, 1 )

( 5, 2 ) ( 4, 3 )( 6, 1 )

( 3, 2, 2 ) ( 3, 3, 1 )

( 4, 1, 1, 1 )

( 3, 2, 1, 1 ) ( 2, 2, 2, 1 )

( 3, 1, 1, 1, 1 )

( 2, 1, 1, 1, 1, 1 )

( 2, 2, 1, 1, 1 )

n = 7

Game TreesGame Trees


(Nim Game) n , ,

, ,

n = 5, , ,

,5

level 12

3

1

1

level 2

3

2 1

4

12

2

1

00 level 3

1 1

1

1

0

1

level 4

123

2 1 1 0

001

level 5

1 2

3

1 2

0

1 1

1

1

1

1

level 0


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5

level 12

3

1

1

level 2

3

2 1

4

12

2

1

00 level 3

1 1

1

1

0

1

level 4

123

2 1 1 0

001

level 5

1 2

3

1 2

0

1 1

1

1

1

1

level 0


Example 5 (P. 653)

Rule: (1) In each step, remove one or more stone from

exactly one of the piles. (2) A player without a legal

move loses.


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Definition 1 (P. 654): The value of a vertex in a game

tree is defined recursively as follows:

(i) The value of a leaf is the payoff to the first palyer

when the game terminates in the position represented

by this leaf.(ii) The value of an internal vertex at an even level is the

maximum values of its children, and the value of an

internal vertex at the odd level is the minimum of

values of its children.

Theorem 3 (P. 654): The value of a vertex of a game

tree tells us the payoff to the first player if both players

follow minimax strategy and paly start from the positionrepresented by this vertex.


Example 7 (P. 655)


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Example 6 (P. 653): Tic-tac-toe



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9.3: Tree Traversal9.3: Tree TraversalProcedures for traversing all vertices of anProcedures for traversing all vertices of anordered rooted treeordered rooted tree

Universal address systemsUniversal address systems

Traversal algorithmsTraversal algorithmsDepthDepth--first traversal:first traversal:

Preorder traversalPreorder traversal

InorderInordertraversaltraversal

PostorderPostordertraversaltraversal

BreadthBreadth--first traversalfirst traversal

Infix/prefix/postfix notationInfix/prefix/postfix notation


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Universal Address SystemsUniversal Address Systems

Example 1Example 1 (P. 660): Display the(P. 660): Display the labelingslabelings of theof the

universal address system. The lexicographic ordering ofuniversal address system. The lexicographic ordering of

thethe labelingslabelings isis

0 < 1.1 < 1.2 < 1.3 < 2 < 30 < 1.1 < 1.2 < 1.3 < 2 < 3

< 3.1< 3.1.1< 3.1.2< 3.1.2.1< 3.1< 3.1.1< 3.1.2< 3.1.2.1

< 3.1.2.2< 3.1.2.3


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Example 2Example 2 (P. 662): List the order of elements by using(P. 662): List the order of elements by using

aapreorder traversalpreorder traversal..

a b e j k n o p f c d g l m h ia b e j k n o p f c d g l m h i

Priorities of

Preorder Traversal

Write, Left, Right


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aapreorder traversalpreorder traversal..


InorderInorder TraversalTraversal

Definition 2Definition 2 (P. 662): Let(P. 662): Let TTbe a ordered rooted tree with rootbe a ordered rooted tree with root rr. If. If

TTconsists onlyconsists only rr, then, then rris theis the inorderinordertraversaltraversalofofTT. Otherwise,. Otherwise,

suppose thatsuppose that TT11,, TT22, . . . ,, . . . , TTnn are theare the subtreessubtrees atat rrfrom left to rightfrom left to right

inin TT. The. The inorderinordertraversaltraversalbegins by visitingbegins by visiting rr. It continues by. It continues by

traversingtraversing TT11

inin inorderinorder, then, then TT22

inin inorderinorder, and so on, until, and so on, until TTnn

isis

traversed intraversed in inorderinorder..

Priorities of Inorder

Traversal

Left, Write, Right


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aa inorderinordertraversaltraversal..


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PostorderPostorder TraversalTraversal

Definition 3Definition 3 (P. 664): Let(P. 664): Let TTbe a ordered rooted tree with rootbe a ordered rooted tree with root rr. If. If

TTconsists onlyconsists only rr, then, then rris theis thepostorderpostordertraversaltraversalofofTT. Otherwise,. Otherwise,

suppose thatsuppose that TT11

,, TT22

, . . . ,, . . . , TTnn

are theare the subtreessubtrees atat rrfrom left to rightfrom left to right

inin TT. The. Thepostorderpostordertraversaltraversalbegins by visitingbegins by visiting rr. It continues by. It continues by

traversingtraversing TT11 inin postorderpostorder, then, then TT22 inin postorderpostorder, and so on, until, and so on, until

TTnn is traversed inis traversed in postorderpostorder..

Priorities of Posorder

Traversal

Left, Right, Write



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aapostorderpostordertraversaltraversal..

Priorities of Posorder

Traversal

Left, Right, Write



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9.4: Spanning Trees9.4: Spanning Trees

Definition 1 (P. 675): Let Gbe a simple graph. A spanning

tree ofG is a subgraph ofG that is a tree containing everyvertex ofG.

not unique


Example 1Example 1 (P. 675): Find a spanning tree of the simple(P. 675): Find a spanning tree of the simple

graphgraph GG..


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Theorem 1Theorem 1 (P. 676): A simple graph is connected if and(P. 676): A simple graph is connected if and

only if it has a spanning tree.only if it has a spanning tree.

Example 2Example 2 (P. 676):(P. 676): IP MulticastingIP Multicasting


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Depth First SearchDepth First Search

Depth First SearchDepth First Search is also calledis also called backtrackingbacktracking, since, since

it returns to vertices previously visited to add path.it returns to vertices previously visited to add path.

"Never burn your bridges after crossing you may have to backtrack someday"


Example 3Example 3 (P. 678): Use a(P. 678): Use aDepth First SearchDepth First Search to findto find

spanning tree for the graphspanning tree for the graph GG..


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Example 4Example 4 (P. 678): Use a(P. 678): Use aDepth First SearchDepth First Search to findto find




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A maze


BreadthBreadth--First SearchFirst Search


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Example 5Example 5 (P. 680): Use a(P. 680): Use aBreadth First SearchBreadth First Search to findto find




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Exploring a town depth-first

not necessary be unique

Exploring a town using breadth-first search.


Backtracking ApplicationsBacktracking Applications

Example 6Example 6 (P. 682):(P. 682): Graph ColoringGraph Coloring Use backtrackingUse backtracking

(depth first search) to decide whether a graph(depth first search) to decide whether a graph GG can becan be

colored usingcolored using nn colors.colors.


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Example 7Example 7 (P. 683):(P. 683): TheThe nn--Queens ProblemQueens Problem UseUse

backtracking (depth first search) to solve thebacktracking (depth first search) to solve the nn--queensqueens

problem.problem.



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Depth First Search in Directed GraphDepth First Search in Directed Graph

Example 9Example 9 (P.684): Do the(P.684): Do the depth first searchdepth first search forfor

the following directed graph.the following directed graph.

Example 10Example 10 (P.685):(P.685): Web SpidersWeb Spiders


Example 10Example 10 (P.685):(P.685): Web SpidersWeb Spiders To indexTo index

websites, search engines such aswebsites, search engines such as googlegoogle, yahoo, yahoo

systematically explore the web starting at asystematically explore the web starting at a

known site. These search engines use programsknown site. These search engines use programs

called Web spiders (or crawlers or bots) to visitcalled Web spiders (or crawlers or bots) to visit

websites and analyze their contents. Webwebsites and analyze their contents. Web

spiders use both depthspiders use both depth--first searching andfirst searching and

breadthbreadth--first searching to create indices.first searching to create indices.


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9.5: Minimum Spanning Trees9.5: Minimum Spanning TreesDefinition 1Definition 1 (P. 689): A(P. 689): A minimum spanning treeminimum spanning tree in ain a

connected weighted graph is a spanning tree that hasconnected weighted graph is a spanning tree that has

the smallest sum of weights of its edges.the smallest sum of weights of its edges.

Robert Clay Prim

1921-

It may not be unique.



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Prims Algorithms ( 1957)

Prims Algorithm

O

A

C

B

E

DT

22

1

5

3 1

7 nodesMST has (7-1) edges.

19


O

O

A

2

Prims Algorithm

O

A

B

22

O

A

C

B

2 2

1

O

A

C

B

E

22

13

O

A

C

B

E

D

22

13

1

O

A

C

B

E

DT

22

1

5

31

20


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Example 1Example 1 (P.690): Find the minimum spanning tree by(P.690): Find the minimum spanning tree by

usingusing PrimPrimss algorithm.algorithm.



usingusing PrimPrimss algorithm.algorithm.


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Joseph BernardKruskal 1928-


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5-edges

6-nodes


Kruskals Algorithm (1956)BC(1) DE(1) AB(2) AO(2) BE(3) BD(4) CE(4) CO(4) BO(5) DT(5) AD(7) ET(7)

O

A

C

B

E

DT

O

A

C

B

E

DT

1 1

O

A

C

B

E

DT

1

add BC add DE


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usingusing KruskalKruskalss algorithm.algorithm.

discrete math- tree

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