Árboles Binarios (UQ)

7/29/2019 rboles Binarios (UQ)

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BinaryTreesAlgorithmsandDataStructures

COMP3506/7505


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BinaryTrees

BinaryTreeDefinitionPropertiesofProperBinaryTrees

BinaryTreeADT InorderTraversal


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BinaryTreeDefinition

Abinarytreeisatreewiththefollowingproperties:

Eachinternalnodehasatmosttwochildren(exactlytwofor

properbinarytrees)

Anodeschildrenareanorderedpair

Aninternalnodehasaleftchildandarightchild

A

B C

F GD E

H I

Applications: arithmeticexpressions decisionprocesses searching


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BinaryTreeRecursiveDefinition

Abinarytreeiseither:

atreeconsistingofasinglenode,or

atreewhoseroothasanorderedpairofchildren,

eachofwhichisabinary

tree

A

B C

F GD E

H I


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ArithmeticExpressionTree

Binarytreeassociatedwithanarithmeticexpression internalnodes:operators externalnodes:operands

Example:arithmeticexpressiontreefortheexpression(2(a-1)+(3b))

+

-2

a 1

3 b


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Exercise:ArithmeticExpressionTree

Generatethebinarytreeassociatedwiththearithmeticexpression

(a

(4-b)+(7

c))


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Exercise:ArithmeticExpressionTree

Generatethebinarytreeassociatedwiththearithmeticexpression

(a

(4-b)+(7

c))+

-a

4 b

7 c

Count the following: Nodes Roots Leaves External nodes Internal nodes

Whats the height of thetree?

Whats the depth of thetree?


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Examplesofarithmeticexpressiontrees

(2(a-1)+(3b))

2a-1+3b

(2a-(1+3)b)

+

-2

a 1

3 b


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(2(a-1)+(3b))

2a-1+3b

(2a-(1+3)b)

+

-

2 a

1 3 b


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(2(a-1)+(3b))

2a-1+3b

(2a-(1+3)b) +

-

2 a

1 3

b


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DecisionTree

Binarytreeassociatedwithadecisionprocess internalnodes:questionswithyes/noanswer externalnodes:decisions

Example:diningdecision

Want a fast meal?

How about coffee? On expense account?

Starbucks Spikes Al Forno Caf Paragon

Yes No

Yes No Yes No


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PropertiesofProperBinaryTrees

Notationn numberofnodes

e numberofexternalnodes

i numberofinternalnodes

h height

Properties: e=i+1 n=2e-1 hi h(n-1)/2 e2h hlog2e hlog2(n+1)1

Level

0

1

2


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BinaryTreeADT

TheBinaryTreeADTextendstheTreeADT,

i.e.,itinheritsallthe

methodsoftheTreeADT Additionalmethods:

positionleft(p) positionright(p) booleanhasLeft(p) booleanhasRight(p)

Updatemethodsmaybedefinedbydata

structuresimplementing

theBinaryTreeADT


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LinkedStructureforBinaryTrees

Anodeisrepresentedbyanobjectstoring

Element Parentnode Leftchildnode Rightchildnode

NodeobjectsimplementthePositionADT

B

DA

C E

B

A D

C E


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Array-BasedRepresentationofBinaryTrees

Nodesarestoredinanarray

letrank(node)bedefinedasfollows: rank(root)=1 ifnodeistheleftchildofparent(node),

rank(node)=2*rank(parent(node))

ifnodeistherightchildofparent(node),rank(node)=2*rank(parent(node))+1

1

2 3

6 74 5

10 11

A

HG

FE

D

C

B

J


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InorderTraversal

Inaninordertraversalanodeisvisitedafteritsleftsubtree

andbeforeitsrightsubtree

Application:Drawabinarytree

x(v)=in-orderrankofv y(v)=depthofv

3

1

2

5

6

7 9

8

4

Algorithm inOrder(v)

ifhasLeft(v) then

inOrder(left(v))

visit(v)

ifhasRight(v) then

inOrder(right(v))


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Whatorderwouldyouvisitthenodesinthisbinarytreeusinganinordertraversal?


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Whatorderwouldyouvisitthenodesinthisbinarytreeusinganinordertraversal?


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PrintArithmeticExpressions

Specializationofaninordertraversal

printoperandoroperatorwhenvisitingnode

print(beforetraversingleftsubtree

print)aftertraversingrightsubtree

+

-2

a 1

3 b((2 (a - 1)) + (3 b))

AlgorithmprintExpression(v)

ifhasLeft(v) thenprint(()

printExpression (left(v))

print(v.element())

ifhasRight(v) thenprintExpression (right(v))

print())


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EvaluateArithmeticExpressions

Specializationofapostordertraversal

recursivemethodreturningthevalueofasubtree

whenvisitinganinternalnode,combinethevaluesofthesubtrees

Algorithm evalExpr(v)

ifisExternal(v) then

returnv.element()

else

x evalExpr(leftChild(v))

y evalExpr(rightChild(v))operator stored at vreturnx y

+

-2

5 1

3 2


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EulerTourTraversal

Generictraversalofabinarytree Includesaspecialcasesthepreorder,postorderandinordertraversals Walkaroundthetreeandvisiteachnodethreetimes:

ontheleft(preorder)

frombelow(inorder)

ontheright(postorder)+

-2

5 1

3 2

L

B

R


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WhatorderwouldyouvisitthenodesinthisbinarytreeusinganEulertourtraversal?


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WhatorderwouldyouvisitthenodesinthisbinarytreeusinganEulertourtraversal?


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TemplateMethodPattern

Genericalgorithmthatcanbespecializedbyredefiningcertainsteps

ImplementedbymeansofanabstractJavaclass

Visitmethodsthatcanberedefinedbysubclasses


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TemplateMethodPattern

Visitmethodsthatcanberedefinedbysubclasses

TemplatemethodeulerTour

Recursivelycalledontheleftandrightchildren

AResultobjectwithfieldsleftResult,rightResultand

finalResultkeepstrackoftheoutputoftherecursivecallstoeulerTour

public abstract class EulerTour {

protected BinaryTree tree;

protected void visitExternal(Position p, Result r) {}

protected void visitLeft(Position p, Result r) {}

protected void visitBelow(Position p, Result r) {}

protected void visitRight(Position p, Result r) {}

protected Object eulerTour(Position p) {

Result r = new Result();

if tree.isExternal(p) {

visitExternal(p, r);

} else {

visitLeft(p, r);

r.leftResult = eulerTour(tree.left(p));

visitBelow(p, r);

r.rightResult = eulerTour(tree.right(p));

visitRight(p, r);

}

return r.finalResult;

}

}


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Evaluating an artihmetic expression


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SpecializationsofEulerTour

SpecializeEulerTourtoevaluateanarithmetic

expression

Assumptions ExternalnodesstoreIntegerobjects Internalnodesstore

Operatorobjectssupportingmethod

operation(Integer,

Integer)

public class EvaluateExpression extends EulerTour{

protected void visitExternal(Position p, Result r){

r.finalResult = (Integer) p.element();}

protected void visitRight(Position p, Result r){

Operator op = (Operator) p.element();r.finalResult = op.operation(

(Integer) r.leftResult, (Integer) r.rightResult );}

}


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LectureSummary

BinaryTreeDefinition PropertiesofProperBinaryTrees BinaryTreeADT

Implementedusinglinksorarray Traversals

InorderEulertour

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