lecture 15 trees and tree traversals -...

CSC212 Data Structure - Section FG

Lecture15TreesandTreeTraversals

Instructor:FengHUDepartmentofComputerScience

CityCollegeofNewYork

Motivation

• Linearstructures• arrays• dynamicarrays• linkedlists

• NonlinearStructures• trees- HierarchicalStructures• Graphs

• Why???

Application:MailingAddresses

Feng HU, CS Dept, CCNY, New York, NY 10031, USA

6 billion = 6,000,000,000 people in the world

What kind of structure is the best for a postman to locate me?

Array ?

Linked list ?

Tree ?

ATreeforallthemailingaddresses

China

Earth

USA... ...Korea

NY... ...

... ...

NYC

MA ... ...

CCNY

F. HU

Albany

... ...... ...

... ...

... ...

CS

TreesandBinaryTrees

•Chapter10introducestrees.• Thispresentationillustratesbasicterminologyforbinarytrees

• andfocuseson• CompleteBinaryTrees:thesimplestkindoftrees

• BinaryTreeTraversals:anykindofbinarytrees

Data Structuresand Other ObjectsUsing C++

BinaryTrees

•Abinarytreehasnodes,similartonodesinalinkedliststructure.

•Data ofonesortoranothermaybestoredateachnode.

•Butitistheconnections betweenthenodeswhichcharacterizeabinarytree.

BinaryTrees

•Abinarytreehasnodes,similartonodesinalinkedliststructure.

•Data ofonesortoranothermaybestoredateachnode.

•Butitisthe connections betweenthenodeswhichcharacterizeabinarytree.

An example canillustrate how theconnections work

ABinaryTreeofStates

Inthisexample,thedatacontainedateachnodeisoneofthe50states.

Washington

Colorado

Oklahoma

Arkansas

Mass.

New

Ham

pshi

re

Arizona

Nebraska

ABinaryTreeofStates

Eachtreehasaspecialnodecalleditsroot,usuallydrawnatthetop.

Washington Arkansas Colorado Florida Oklahoma Arizona Mass. NewHampshire Nebraska

ABinaryTreeofStates

Eachtreehasaspecialnodecalleditsroot,usuallydrawnatthetop. The example tree

has Washingtonas its root.


ABinaryTreeofStates

Eachnodeispermittedtohavetwolinkstoothernodes,calledtheleftchild andtherightchild.


ABinaryTreeofStates

Childrenareusuallydrawnbelowanode.

The right child ofWashington is

Colorado.

The left child ofWashington is

Arkansas.


ABinaryTreeofStates

Somenodeshaveonlyonechild.

Arkansas has aleft child, but no

right child.


AQuiz


Which node hasonly a right child?


AQuiz


Florida hasonly a right child.


ABinaryTreeofStates

Anodewithnochildreniscalledaleaf.


ABinaryTreeofStates

Eachnodeiscalledtheparent ofitschildren.

Washington is theparent of Arkansas

and Colorado.Washington Arkansas Colorado Florida Oklahoma Arizona Mass. New

Hampshire Nebraska

ABinaryTreeofStates

Tworulesaboutparents:

❶ The root has no parent.

❷ Every other node has exactly one parent.


ABinaryTreeofStates

Twonodeswiththesameparentarecalledsiblings.

Arkansasand Coloradoare siblings.


CompleteBinaryTrees

Acompletebinarytreeisaspecialkindofbinarytreewhichwillbeusefultous.

CompleteBinaryTrees

Acompletebinarytreeisaspecialkindofbinarytreewhichwillbeusefultous.

When a completebinary tree is built,

its first node must bethe root.

CompleteBinaryTrees

Thesecondnodeofacompletebinarytreeisalwaystheleftchildoftheroot...

CompleteBinaryTrees

Thesecondnodeofacompletebinarytreeisalwaystheleftchildoftheroot......andthethirdnodeisalwaystherightchildoftheroot.

CompleteBinaryTrees

Thenextnodesmustalwaysfillthenextlevelfromlefttoright.

IsThisComplete?

IsThisComplete?

Yes!✔ Itiscalledtheemptytree,andit

hasnonodes,notevenaroot.

FullBinaryTrees

Afullbinarytreeisaspecialkindofcompletebinarytree

When a fullbinary tree is built,

its first node must bethe root.

FULL

FullBinaryTrees

Thesecondnodeofafullbinarytreeisalwaystheleftchildoftheroot...

not FULL yet

FullBinaryTrees

Thesecondnodeofafullbinarytreeisalwaystheleftchildoftheroot......andyouMUSThavethethirdnodewhichalwaystherightchildoftheroot. FULL

FullBinaryTrees


not FULL yet

FullBinaryTrees

Thenextnodesmustalwaysfillthenextlevelfromlefttoright...untileveryleafhasthesamedepth(2)

FULL!

FullBinaryTrees


IsThisFull?

IsThisFull?

Yes!✔ Itiscalledtheemptytree,andit

hasnonodes,notevenaroot.

ImplementingaCompleteBinaryTree

•Wewillstorethedatefromthenodesinapartially-filledarray.

An array of dataWe don't care what's in

this part of the array.

An integer to keeptrack of how many nodes are in the tree

3

ImplementingaCompleteBinaryTreeUsinganArray

•Wewillstorethedatefromthenodesinapartially-filledarray.

An array of dataWe don't care what's in

this part of the array.

An integer to keeptrack of how many nodes are in the tree

3

Read Section 10.2 tosee details of how

the entries are stored.

ImplementingaCompleteBinaryTreeUsinganArray

• Rootisatcomponent[0]• Parentofnodein[i]isat[(i-1)/2)• Children(ifexist)ofnode[i]isat[2i+1]and[2i+2]

• Totalnodenumber• 20+21+22+…+2d-1+r, r<=2d,disthedepth

BinaryTreeSummary

•Binarytreescontainnodes.• Eachnodemayhavealeftchildandarightchild.• Ifyoustartfromanynodeandmoveupward,youwilleventuallyreachtheroot.

• Everynodeexcepttheroothasoneparent.Theroothasnoparent.

•Completebinarytreesrequirethenodestofillineachlevelfromleft-to-rightbeforestartingthenextlevel.

BinaryTreeBasics

Abinarytreeisastructureinwhich:

Eachnodecanhaveatmosttwochildren,andinwhichauniquepathexistsfromtheroottoeveryothernode.

Thetwochildrenofanodearecalledtheleftchild andtherightchild, iftheyexist.

ABinaryTreeExercise

Q

V

T

K S

AE

L

Howmanyleafnodes?

Q

V

T

K S

AE

L

HowmanydescendantsofQ?

Q

V

T

K S

AE

L

HowmanyancestorsofK?

Q

V

T

K S

AE

L

Question: How to implement a general binary tree ?

ImplementingaBinaryTreewithaClassforNodes

Q

V

T

K S

AE

L

Root

BinaryTreeNodes

• Eachnodeofabinarytreeisstoredinanobjectofanewbinary_tree_nodeclassthatwearegoingtodefine

• Eachnodecontainsdataaswellaspointerstoitschildren(nodes)• Anentiretreeisrepresentedasapointertotherootnode

binary_tree_nodeClass

• variables• functions

template <class Item>class binary_tree_node{public:

......private:

Item data_field;binary_tree_node *left_field;binary_tree_node *right_field;

};

bintree

//retrievalsdataleftright//setset_dataset_leftset_right//booleanis_leaf

CreatingandManipulatingTrees

• Consideronlytwofunctions• Clearingatree

• Returnnodesofatree totheheap• Copyingatree

• TheImplementationiseasierthanitseems• ifweuserecursivethinking

ClearingaTree

Q

V

T

K S

AE

L

Root

Clear LEFT SUBTREE

ClearingaTree

V

S

A

L

Root

Clear RIGHT SUBTREE

ClearingaTree

VRoot

Return root node to the heap

ClearingaTree

NULL Root

Set the root pointer to NULL

ClearaTree

• key:recursivethinking

template <class Item>void tree_clear(binary_tree_node<Item>*& root_ptr)// Library facilities used: cstdlib{

if (root_ptr != NULL){

tree_clear( root_ptr->left( ) ); // clear left sub_treetree_clear( root_ptr->right( ) ); // clear right sub_treedelete root_ptr; // return root node to the heaproot_ptr = NULL; // set root pointer to the null

}}

bintree

CopyaTree

• Canyouimplementthecopy?(p467)

template <class Item>binary_tree_node<Item>* tree_copy(const binary_tree_node<Item>* root_ptr)// Library facilities used: cstdlib{

binary_tree_node<Item> *l_ptr;binary_tree_node<Item> *r_ptr;

if (root_ptr == NULL)return NULL;

else{

l_ptr = tree_copy( root_ptr->left( ) ); // copy the left sub_treer_ptr = tree_copy( root_ptr->right( ) ); // copy the right sub_treereturn

new binary_tree_node<Item>( root_ptr->data( ), l_ptr, r_ptr);} // copy the root node and set the the root pointer

}

bintree

BinaryTreeTraversals

• pre-order traversal• root(leftsub_tree) (right sub_tree)

• in-ordertraversal• (leftsub_tree) root(right sub_tree)

• post-order traversal• (leftsub_tree) (right sub_tree)root

• backwardin-ordertraversal• (right sub_tree)root (leftsub_tree)

bintree

PreorderTraversal:JEAHTMY

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print left subtree second Print right subtree last

Print first

PreorderTraversal

• Example:printthecontentsofeachnode

template <class Item>void preorder_print(const binary_tree_node<Item>* node_ptr)// Library facilities used: cstdlib, iostream{

if (node_ptr != NULL){

std::cout << node_ptr->data( ) << std::endl;preorder_print(node_ptr->left( ));preorder_print(node_ptr->right( ));

}}

InorderTraversal:AEHJMTY

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print left subtree first Print right subtree last

Print second

InorderTraversal


template <class Item>void inorder_print(const binary_tree_node<Item>* node_ptr)// Library facilities used: cstdlib, iostream{


inorder_print(node_ptr->left( ));std::cout << node_ptr->data( ) << std::endl;inorder_print(node_ptr->right( ));

}}

Postorder Traversal: A H E M Y T J

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print left subtree first Print right subtree second

Print last

PostorderTraversal


template <class Item>void postorder_print(const binary_tree_node<Item>* node_ptr)// Library facilities used: cstdlib, iostream{


postorder_print(node_ptr->left( ));postorder_print(node_ptr->right( ));std::cout << node_ptr->data( ) << std::endl;

}}

BackwardInorderTraversal:YTMJHEA

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print right subtree firstPrint left subtree last

Print second

BackwardInorderTraversal:YTMJHEA

Print right subtree first

Print left subtree last

Print second

AUsefulBackwardInorderTraversal

• Intenteachnumberaccordingitsdepth

template <class Item, class SizeType>void print(binary_tree_node<Item>* node_ptr, SizeType depth)// Library facilities used: iomanip, iostream, stdlib{


print(node_ptr->right( ), depth+1);std::cout << std::setw(4*depth) << ""; // Indent 4*depth spaces.std::cout << node_ptr->data( ) << std::endl;print(node_ptr->left( ), depth+1);

}}

bintree

AChallengingQuestion:

• Forthetraversalswehaveseen,the“processing”wassimplyprintingthevaluesofthenode

•Butwe’dliketodoanykindofprocessing• Wecanreplace“cout”withsomeotherformof“processing”

•Buthowabout1000kinds?• Cantemplatebehelpful?

• Solution::::::::> (pages501– 507)

Aparametercanbeafunction

•writeonefunctioncapableofdoinganything•Aparametertoafunctionmaybeafunction.Suchaparameterisdeclaredby• thenameofthefunction’sreturntype(orvoid),• thenthenameoftheparameter(i.e.thefunction),• andfinallyapairofparentheses().• Inside()isalistofparametertypesofthatparameterfunction

• Example• intsum(voidf(int&,double),inti,...);

PreorderTraversal– printonly


template <class Item>void preorder_print(const binary_tree_node<Item>* node_ptr)// Library facilities used: cstdlib, iostream{


std::cout << node_ptr->data( ) << std::endl;preorder_print(node_ptr->left( ));preorder_print(node_ptr->right( ));

}}

PreorderTraversal– generalform

• Atemplatefunctionfortreetraversals

template <class Item>void preorder(void f(Item&), binary_tree_node<Item>* node_ptr)// Library facilities used: cstdlib{


f( node_ptr->data( ) ); // node_ptr->data() return reference !preorder(f, node_ptr->left( ));preorder(f, node_ptr->right( ));

}}

PreorderTraversal– howtouse

• Definearealfunctionbeforecalling

void printout(int & it)// Library facilities used: iostream

{std::cout << it << std::endl;

}

Can you print out all the node of a tree pointed by root ?

binary_tree_node<int> *root; ....preorder(printout, root); Yes!!!

PreorderTraversal– anotherfunctions

• Candefineotherfunctions...

void assign_default(int& it)// Library facilities used: iostream

{it = 0;

} // unfortunately template does not work here for function parameters

You can assign a default value to all the node of a tree pointed by root:

binary_tree_node<int> *root; ....preorder(assign_default, root);


• Canthefunction-argumentsbetemplate?

template <class Item>void printout(Item& it)// Library facilities used: iostream


}


binary_tree_node<string> *root; ....preorder(print_out, root); X ! print_out should have real types


• Thefunction-argumentsmaybetemplateif...

template <class Item>void printout(Item& it)// Library facilities used: iostream


}


binary_tree_node<string> *root; ....preorder(print_out<string>, root);

But you may do the instantiation like this

PreorderTraversal– amoregeneralform

• Anextremelygeneralimplementation(p505)

template <class Process, class BTNode>void preorder(Process f, BTNode* node_ptr)

// Note: BTNode may be a binary_tree_node or a const binary tree node.// Process is the type of a function f that may be called with a single// Item argument (using the Item type from the node),// as determined by the actual f in the following.// Library facilities used: cstdlib

{if (node_ptr != NULL){

f( node_ptr->data( ) );preorder(f, node_ptr->left( ));preorder(f, node_ptr->right( ));

}}

bintree

FunctionsasParameters

• WecandefineatemplatefunctionX withfunctionsasparameters– whicharecalledfunctionparameters

• AfunctionparametercanbesimplywrittenasProcessf(whereProcessisatemplate),andtheformsandnumberofparametersforf aredeterminedbytheactualcalloffinsidethetemplatefunctionX

• TherealfunctionargumentforfwhencallingthethetemplatefunctionX cannotbeatemplatefunction,itmustbeinstantiatedinadvanceorrightinthefunctioncall

Summary

• Tree,BinaryTree,CompleteBinaryTree• child,parent,sibling,root,leaf,ancestor,...

•ArrayRepresentationforCompleteBinaryTree• Difficultifnotcompletebinarytree

•AClassofbinary_tree_node• eachnodewithtwolinkfields

• TreeTraversals• recursivethinkingmakesthingsmucheasier

•AgeneralTreeTraversal• AFunctionasaparameterofanotherfunction

THE END

Presentation copyright 1997 Addison Wesley Longman,For use with Data Structures and Other Objects Using C++by Michael Main and Walter Savitch.

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lecture 15 trees and tree traversals -...

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