Agraph - - PDF document

• Agraph G=(V,E)consistsofansetVofVERTICES

andasetEofedges,withE={(u,v):u,v∈V,u≠ v Atree isaconnectedgraphwithnocycles. ∃ apathbetweeneachpairofvertices.

• WhatisaTree
• Abstractmodelofa

hierarchicalstructure

• Atreeconsistsof

nodeswithaparent) childrelation

• Applications:

– Organizationcharts – Filesystems – Programming environments

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TreeTerminology

• Root:nodewithoutparent(A)
• (
• ,
• !

.

• Descendant ofanode:child,grandchild,grand)grandchild,etc.
• Internalnode:nodewithatleast
• nechild(A,B,C,F)
• Ancestors ofanode:parent,

grandparent,grand)grandparent,etc.

• Externalnode (a.k.a.leaf ):

nodewithoutchildren(E,I,J,K,G,H,D)

• Subtree:treeconsisting
• fanodeandits

descendants

• Depth ofanode:numberof

ancestors(=distancefromtheroot)

• Height ofatree:maximumdepthof

anynode(3)

TreeTerminology

• (
• ,
• !

.

Distancebetweentwonodes:number

• f“edges” betweenthem

• genericcontainermethods

) size(),isEmpty(),elements()

• positionalcontainermethods

) positions(),swapElements(p,q),replaceElement(p,e)

• querymethods

) isRoot(p),isInternal(p),isExternal(p)

• accessor methods

) root(),parent(p),children(p)

• updatemethods

) applicationspecific

• Algorithmdepth(T,v)

ifT.isRoot(v)then return0 else return1+depth(T,T.parent(v))

• Ifvistherootthedepthis0

Ifvisaninternalnodethedepthis1+thedepthofitsparent Complexity?

• Atraversalvisitsthenodesof

atreeinasystematicmanner

• Inapreordertraversal,a
• Application:printastructured

document

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1 % 5 6 7 8 9 : ;

• DBACFEHLIG
• Inapostordertraversal,a
• Application:computespace

usedbyfilesinadirectory anditssubdirectories

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&=< 2> 1. < 2?3 1@. 2?3 %6. &12 5. &12 %. +2?3 %@.

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ACBFLHIGED

• Letd(x)bethenumber
• fsub)treesofnodex.

Start:x=root IN)ORDERVISIT 1. Visitthefirstsub)tree(inorder) 2. Visittheroot 3. Visitthesecondsub)tree(inorder) M M d(x)+1. Visitthed(x)th sub)tree(inorder)

Inorder !"

ABCDFLHEIG

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PhilipIII, Charles,

WhenCharlesdies,PhilipIIbecomesKing. IfPhilipIIdiesaswell….

PhilipII, CharlesI, RudolphII,Ernest,Mathias,Max,Albert,Wenzel,

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rightchild leftchild

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isaleaf,or hastwochildren

{

Eachnode: Fullbinarytreeswithallleavesatthesame level: #

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• fdepthh=Perfecttreesofdepth(h)1)

+

• neormoreleavesatlevelh.

Leavesgoattheleft

• Notationforbinarytree.

Inthebook: childrenare“completed” with“fake” nodes Thegreensquarednodesarethedummynodes. InthiswayALLtheoriginalnodesareinternal. Theleavesarethefakegreennodes. AlltreesareFULL

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rightchild leftchild

• DecisionTree
• Binarytreeassociatedwithadecisionprocess

– internalnodes:questionswithyes/noanswer – externalnodes:decisions

• Example:diningdecision

)A =+A ">A + B

• C

D # D # D #

#

• ArithmeticExpressionTree

%

– internalnodes:operators – externalnodes:operands

#

+ × × − %

• 1

5 + Example:arithmeticexpression treefortheexpression (2× (a) 1)+(3× b))

• #
• Notation
• #ofnodes

#ofleaves

• #ofinternalnodes

height Maximumnumberof nodesateachlevel?

Level0 Level1 1 2 4 8 Level2 leveli))))))) 2i

• #
• Notation

numberofnodes numberofleaves

• numberof

internalnodes height

• Properties:

– ( ' 1 – ( 2! 1 – ≤ ≤ ≤ ≤ – ≤ ≤ ≤ ≤ (! 1))2 – ≤ ≤ ≤ ≤ 2 – ≥ ≥ ≥ ≥ log2 – ≥ ≥ ≥ ≥ log2 (' 1) ! 1

• ( ' "
• ( ' "
• ( ' "
• ( 2! 1

( + ( *' ( ' 1(justproved) (!+

• ≤

≤ ≤ ≤ (h=maxn.ofancestors) (,- (. Ex:h=3,i=3

Theremustbeatleastoneinternalnodeforeachlevel (exceptthelast)!

• ≤

≤ ≤ ≤ #

leveli))))))) maxn.ofnodesis2i h=3 23 leaves ifallatlast levelh

• therwiseless
• /≤

≤ ≤ ≤ #

≥ ≥ ≥ ≥ log2 log2 ≤

≤ ≤ ≤ log2 #

log2 ≤

≤ ≤ ≤ h

• &'

withheighththereare 2h+1 )1nodes Ateachlevelthereare2l nodes,sothetreehas:

h

∑ 2l =1+2+4+… +2h =2h+1)1

• ( 2h+1 )1
• l=0
• InBinarytrees:

log(n+1)≤

≤ ≤ ≤ h+1

h≥

≥ ≥ ≥ log(n+1))1

Asaconsequence: n≤

≤ ≤ ≤ 2h+1)1

n+1≤

≤ ≤ ≤ 2h+1

• bviously ≤

≤ ≤ ≤ 2h+1 )1

• 0&#1

with heighth ≤ " ≤ #$ Frompreviousobservation:n≤ ≤ ≤ ≤ 2h+1) 1 n≥

≥ ≥ ≥ 2h

• Acompletebinarytreeisaperfectbinarytreeofheight

h)1plussomemoreleaves… 2h) 1

• Height ofa with

nnodes:  %" 02 n≥

≥ ≥ ≥ 2h

• #
• accessor methods

)leftChild(p),rightChild(p),sibling(p)

• updatemethods

)expandExternal(p),removeAboveExternal(p)

• therapplicationspecificmethods
• Pre),post),in) (order)
• Refertotheplaceoftheparent

relativetothechildren

• pre isbefore:parent,child,child
• post isafter:child,child,parent
• in

isinbetween:child,parent,child

#

• #

Preorder,Postorder,

3(T,) ()

• 3 (-45& ")

3 (-46 & ") 3(T,)

• 3 (-45& ")

3(-46 & ") ()

• Inorder

!"

3(T,)

• 3 (-45& ")

()

• 3(-46 & ")

#

• −
• +

ArithmeticExpressions

Inorder:a– b Postorder:ab– Preorder– ab + × × − %

• 1

5 + %× − 1+ 5× + Inorder: %1E × 3 +× + Postorder:

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+,-$,.-,/ #&*) ( '

• +01$,.-,/

&'( ) *# 23$,.-,/ &#'( ) *

• EvaluateArithmeticExpressions
• Specializationofa

traversal

– recursivemethod returningthevalueofa subtree – whenvisitinganinternal node,combinethevalues

• fthesubtrees
• ←

← ◊ ◊ ◊ ◊ ← && ◊ ◊ ◊ ◊

+ × × − % 6 1 5 %

• +

× × − % 6 1 5 %

+

Eval × − % 6 1 × 5 % Eval − 1 6 % Eval Eval × 5 % Eval Eval × 6 1 −

• PrintArithmeticExpressions
• Specializationofan

traversal

– printoperandor

• peratorwhenvisiting

node – print“(“ before traversingleftsubtree – print“)“ aftertraversing rightsubtree

FGBB

• FHBB

+ × × − %

• 1

5 + GG%× − 1HH+ G5× +HH %× − 1+ 5× +

• AlgorithmpreOrderTraversalwithStack(T)

StackS TreeNodeN S.push(T)//pushthereferencetoTintheemptystack While(notS.empty()) N=S.pop() if(N!=null){ print(N.elem)//printinformation S.push(N.rightChild)//pushthereferenceto theleftchild S.push(N.leftChild)//pushthereferenceto therightchild }

• Algorithm preOrderTraversalwithStack(T)
• +
• &
• S.push(T)//pushthereferencetoTintheemptystack

N=S.pop() print(N.elem)

• Algorithm preOrderTraversalwithStack(T)
• +
• &
• S.push(T)//pushthereferencetoTintheemptystack

N=S.pop() print(N.elem) N

• Algorithm preOrderTraversalwithStack(T)
• +
• &
• S.push(N.rightChild)//pushthereferenceto

theleftchild

• S.push(N.leftChild)//pushthereferenceto

therightchild

• Algorithm preOrderTraversalwithStack(T)
• +
• &
• N=S.pop()
• Algorithm preOrderTraversalwithStack(T)
• +
• &
• N=S.pop()
• N

print(N.elem)

+

• Algorithm preOrderTraversalwithStack(T)
• +
• &
• +
• S.push(N.rightChild)

S.push(N.leftChild)

• generictraversalofabinarytree
• thepreorder,inorder,andpostorder traversalsare

specialcasesoftheEuler tourtraversal

• “walkaround” thetreeandvisiteachnodethreetimes:

– ontheleft – frombelow – ontheright

• (T,)
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2"

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• Anodeisrepresented

byanobjectstoring

– Element – Parentnode – Leftchildnode – Rightchildnode

• Nodeobjects

implementthePosition ADT

(

• ∅

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ (

• ∅

∅ ∅ ∅

• swapElements(p,q)Input:2PositionsOutput:None

replaceElement(p,e)

Input:PositionandanobjectOutput:Object

isRoot(p)Input:Position

Output:Boolean

isInternal(p)Input:PositionOutput:Boolean isExternal(p)Input:PositionOutput:Boolean leftChild(p),rightChild(p),sibling(p):

Input:PositionOutput:Position

• #8

39 Element #8 left,right,parent

• leftChild(v)returnv.left

rightChild(v)returnv.right sibling(v) p← parent(v) q← leftChild(p) if(v=q)returnrightChild(p) elsereturnq

• replaceElement(v,obj)

temp← v.element v.element← obj returntemp swapElements(v,w) temp← w.element w.element← v.element v.element← temp

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• expandExternal(v):

new1andnew2arethenewnodes ifisExternal(v) v.left← new1 v.right← new2 size← size+2

Otherinterestingmethodsforthe ADTBinaryTree:

expandExternal(v):Transformvfromanexternal nodeintoaninternalnodebycreatingtwonewchildren

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• removeAboveExternal(v):

(

• ,

(

• ,

(

• ,
• removeAboveExternal(v):

ifisExternal(v) {p← parent(v) s← sibling(v) ifisRoot(v)s.parent← nullandroot← s else {g← parent(p) ifpisleftChild(g)g.left← s elseg.right← s s.parent← g } size← size)2}

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• Anodeisrepresented

byanobjectstoring

– Element – Parentnode – Sequenceofchildren nodes

• Nodeobjectsimplement

thePositionADT

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treeT binarytreeT'representingT

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u’ inT’ firstchildofuinTisleftchildofu’ inT’ firstsiblingofuinTisrightchildofu’ inT’ RULES

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B A D I C E G F H L

• RULE:

touinTcorrespondsu’ inT’ IfuisinternalinTandvisitsfirstchild thenv’ istheleftchildofu’ inT’ ifuisaleafinTandhasnosiblings, thenthechildrenofu’ areleaves Ifvhasasiblingwimmediatelyfollowingit, w’ istherightchildofv’ inT’

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