1 Today’s Objectives Announcements Hand in Homework 3 next week Iterators Trees Tree terminology Depth and height Preorder and postorder traversals

1

Today’s Objectives

Announcements• Hand in Homework 3 next week

Iterators Trees

• Tree terminology• Depth and height• Preorder and postorder traversals

Binary Trees• Proper binary tree• Inorder traversal• Implementing a binary tree with linked nodes• Implementing a binary tree with an array list

Week 11

2

Iterators

3

Iterator

An object used to process the elements in a data structure sequentially

An iterator always has a consistent interface for every data structure

• Iterator interface is always the same for all data structures• Client program sets up a loop that uses an iterator to process

each element

Important steps in an iterator’s implementation• Initialize the iteration• Return a copy of the next element• Implementation depends on the data structure that the iterator is

written for

Iterators (Dale, 64, 110)

4

Iterator ADT

hasNext() Test whether there are elements in the iteratorReturns: boolean

next() Return the next element and step to the next position

Returns: E

Iterators (Goodrich, 242)

5

Iterable ADT

iterator() Return an iterator of the elements in a collection


All collection ADTs should implement the interable ADT, so that collection objects will be able to loop through their elements.

To guarantee this, we could make sure the data structure’s

interface extends the java.lang.Iterable interface

public interface PositionList<E> extends Iterable<E>{ public Iterator<E> iterator(); //other methods of the list ADT}

The Iterable interface requires only one method, the iterator() method, which returns an iterator

6

Using an Iterator

As long as a data structure implements the java.lang.Iterable interface, the code shown here can be used to loop through its elements, regardless of which data structure it is


public static void main(String[] args){

NodePositionList<Character> myList =

new NodePositionList<Character>();

myList.addLast('a');

myList.addLast('b');

myList.addLast('c');

Iterator<Character> it = myList.iterator();

while( it.hasNext() ) {

System.out.println( it.next() );

}

}

7

For-Each Loop

The for-each loop is a shorthand way of writing a loop that iterates through all the elements stored in a data structure

However, this works only with data structures that implement the java.lang.Iterable interface


public static void main(String[] args){

NodePositionList<Character> myList =

new NodePositionList<Character>();

myList.addLast('a');

myList.addLast('b');

myList.addLast('c');

for( Character c : myList ) {

System.out.println(c);

}

}

8

Trees

Chapter 7

9

Tree

Container that stores elements hierarchically.

A tree is a set of nodes storing elements in a parent-child relationship.

Trees (Goodrich, 267)

10

British Royal Family TreeTrees

Queen Victoria

King Edward VII

Albert Victor

Alice

King George V

King George VIKing Edward VIII

Queen Elizabeth II

Charles Ann Andrew Edward

William Henry Peter Zara Beatrice Eugenie

11

TerminologyTrees (Goodrich, 267)

Root

12


RootParentChild

13


Root

ParentChild

14


Root

Tree = a set of nodes storing elements in a parent-child relationship

• A special node called the root has no parent node

• Each of the other nodes has a unique parent node

• According to our textbook, a tree may be empty

15

Are these trees?Trees

1 2

34

16

More TerminologyTrees (Goodrich, 268)

Siblings = two or more nodes that are children of the same parent

17



Internal node = node with one or more childrenExternal node = node with no childrenLeaf = external node

18



Subtree rooted at v

vv

Internal node = node with one or more childrenExternal node = node with no childrenLeaf = external node

19

Recursive Structure of TreesTrees

r

t

s

u

v

w x

20


r

t

s

u

v

w x

21


r

t

s

u

v

w x

22


Ancestor = parent of a node or the parent of an ancestor of a node

u

Descendent = child of a node or the child of a descendent of a node

w

3 ancestors of w

5 descendents of u

23

Varieties of Trees

Ordered Tree• The children of each node have a linear order,

e.g. first, second, third, etc.• Order is drawn from left to right• Examples: slide 8 and fig. 7.4

Binary Tree• Every node has at most two children• Child nodes are called “left child” and “right

child”

Trees (Goodrich, 271, 282)

24

Proper Binary Tree

Each node has either two children or no children To make a binary tree into a proper binary tree, we

can add empty nodes as leaves


25

Examples of How Trees are Used

File system – Fig. 7.3, p. 268 Ordered Tree

• Structured document – Fig. 7.4, p. 269• XML documents

Binary Trees• Decision tree – Fig. 7.10, p. 282• Expression tree – Fig. 7.11, p. 283

Trees (Goodrich, 268–269, 282–283)

26

c < b < a

Decision Tree

Represents “yes” or “no” outcomes Internal nodes are questions External nodes show what to do, based on the answers

leading to it


a < b ?

Yes No

b < c ? b < c ?

a < c ?a < b < c

Yes No

Yes No

a < c ?

Yes

Yes No

No

Example: What is the order of the values in the variables a, b, and c?

a < c < b c < a < b b < a < c b < c < a

27

Expression Tree

External nodes are variables or constants Internal nodes are operators, +, -, *, or /


*

3

-

41 2

+

What is the expression represented by this tree?

28

Tree ADT

How are the data related?• Hierarchically

Operations• Accessor operations

pos root()pos parent( pos )iter children( pos )

• Update operationsvoid replace(v,e)


• Query operationsbool isEmpty()int size()bool

isInternal( pos )bool

isExternal( pos )bool isRoot( pos )coll positions()iter iterator()

29

Tree Interface

public interface Tree<E> {

public int size();

public boolean isEmpty();

public Iterator<E> iterator();

public Iterable<Position<E>> positions();

public E replace(Position<E> v, E e)throws InvalidPositionException;

public Position<E> root() throws EmptyTreeException;

public Position<E> parent(Position<E> v)throws InvalidPositionException, BoundaryViolationException;

public Iterable<Position<E>> children(Position<E> v) throws InvalidPositionException;

public boolean isInternal(Position<E> v) throws InvalidPositionException;

public boolean isExternal(Position<E> v) throws InvalidPositionException;

public boolean isRoot(Position<E> v)throws InvalidPositionException;

}


30

Running Time AssumptionsTrees (Goodrich, 272)

O(n)iterator, positions

O(1)replace

O(1)swapElementsO(cv) (cv = no. of children of v)children

O(1)isRoot

O(1)isInternal, isExternal

O(1)parent

O(1)root

TimeMethod

31

Depth of a Node in a Tree

Depth of v = number of ancestors of v• If v is the root, v’s depth = 0• Else, v’s depth = 1 + depth of v’s parent


Depth

0

1 + depth( myParent ) = 1 + 0 = 1

1 + depth( myParent ) = 1 + 1 = 2

1 + depth( myParent ) = 1 + 2 = 3

32

Finding the Depth

Algorithm depth( T, v ) if T.isRoot(v) then return 0 else return 1 + depth( T, T.parent(v) )


33

Finding the Depth

Algorithm depth( T, v ) if T.isRoot(v) then return 0 else return 1 + depth( T, T.parent(v) )


public static <E> int depth(Tree<E> T, Position<E> v){ if( T.isRoot(v) ) return 0; else return 1 + depth( T, T.parent(v) );}

What is the Big-Oh if n = total number of nodes?

34

Height

Height of node v• If v is an external node, v’s height = 0• Else, v’s height = 1 + maximum height of v’s children

Trees (Goodrich, 274–275)

v

hv = 1 + max( h of myChildren ) hv = 1 + 1 = 2

h = 0 (External node)

h = 1 + max( h of myChildren )h = 1 + 0 = 1


35

Height

Height of node v• If v is an external node, v’s height = 0• Else, v’s height = 1 + maximum height of v’s children


v

hv = 1 + max( h of myChildren ) hv = 1 + 1 = 2


Height of tree T• T’s height = height of the root of T

h = 1 + max( h of myChildren )h = 1 + 0 = 1


36

Finding the Height of a Tree

Algorithm height( T, root ) if T.isExternal(root) then return 0 else h = 0 for each node T.children(root) do h = max( h, height(T,node) ) return 1 + h r

u

t

vh = 0h = max(h,height(T,t))return 1 + h

r 0 0

1

2

T


37



u

t


r 0 0

1

2

h = 0h = max(h,height(T,u))h = max(h,height(T,v))return 1 + h

t

T


38



u

t


r 0 0

1

2


t

return 0

return 0u

T


39



u

t


r 0 0

1

2


t

return 0 return 0

return 0ureturn 0v

T


40



u

t


r 0 0

1

2


t

return 0

return 0u

T


return 0

return 0v

return 1

41



u

t


r 0 0

1

2

return 1

return 2


t

return 0 return 0

return 0ureturn 0v

T


42

Traversal

A systematic way of visiting all the nodes in a tree

When we traversed a list, it was clear which directions a traversal could go:

• Forward from the head

• Backward from the tail

A tree has more choices, so we simplify them by always starting at the root.


43

Preorder Traversal

One way to make sure that we visit every node


R

3

b

41 2

a

T

Start by visiting the root v

R

44

Preorder Traversal



R

3

b

41 2

a

T


Then traverse each subtree v

Ra

45

Preorder Traversal



R

3

b

41 2

a

T



v

Ra1

46

Preorder Traversal



R

3

b

41 2

a

T



v v

Ra12

47

Preorder Traversal



R

3

b

41 2

a

T



v v

v

Ra12b

48

Preorder Traversal



R

3

b

41 2

a

T



v v v

v

Ra12b3

49

Preorder Traversal



R

3

b

41 2

a

T



v v v v

v

Ra12b34

50

Preorder Traversal

Algorithm preorder( T, v ) visit v for each child of v do preorder( T, child )


public static <E> String toStringPreorder(Tree<E> T, Position<E> v){ String s = v.element().toString(); for(Position<E> w : T.children(v)) { s += ", " + toStringPreorder(T, w); } return s;}

51

Postorder Traversal

1

Start by traversing each subtree R

3

b

41 2

a

T

v


52

Postorder Traversal

12


3

b

41 2

a

T

v v


53

Postorder Traversal

Then visit the root

12a


3

b

41 2

a

T

v

v v


54

Postorder Traversal

Then visit the root

12a3


3

b

41 2

a

T

v

v v v


55

Postorder Traversal

Then visit the root

12a34


3

b

41 2

a

T

v

v v v v


56

Postorder Traversal

Then visit the root

12a34b


3

b

41 2

a

T

v

v v v v

v


57

Postorder Traversal

Then visit the root

12a34bR


3

b

41 2

a

T

v

v

v v v v

v


58

Postorder Traversal

Algorithm postorder( T, v ) for each child of v do postorder( T, child ) visit v


public static <E> String toStringPostorder(Tree<E> T, Position<E> v){ String s = ""; for(Position<E> w : T.children(v)) { s += toStringPostorder(T, w) + " "; } s += v.element().toString(); return s;}

59

Tree ADT

How are the data related?• Hierarchically

Operations• Accessor operations

pos root()pos parent( pos )iter children( pos )

• Update operationsvoid replace(v,e)


• Query operationsbool isEmpty()int size()bool

isInternal( pos )bool

isExternal( pos )bool isRoot( pos )coll positions()iter iterator()

60

Binary Tree ADT

The Binary Tree is a specialization of the Tree ADT

Each node has either two children or no children

Child nodes are called “left child” and “right child”

Additional Operations:• Accessor operations

pos left( pos )pos right( pos )pos hasLeft( pos )pos hasRight( pos )

Trees (Goodrich, 282, 284)

61

Binary Tree Interface

public interface BinaryTree<E> extends Tree<E> {

public Position<E> left(Position<E> v)throws InvalidPositionException, BoundaryViolationException;

public Position<E> right(Position<E> v)throws InvalidPositionException, BoundaryViolationException;

public boolean hasLeft(Position<E> v)throws InvalidPositionException;

public boolean hasRight(Position<E> v) throws InvalidPositionException;

}


62

Linked Binary Tree

The Linked Binary Tree implements the Binary Tree ADT

It has all the Binary Tree interface methods, as well as all of the Tree interface methods

Additional operations:• Accessor operations

pos sibling( pos )


• Update operationspos addRoot( e )pos insertLeft( pos, e )pos insertRight( pos, e )e remove( pos )void attach(pos, T1, T2)

63

Proper Binary Tree

Each node has either two children or no children To make a binary tree into a proper binary tree, we

can add empty nodes as leaves


64

Properties of a Proper Binary Tree

Level = all nodes at the same depth Level d has at most 2d nodes


Level

1

0

2

Max Nodes

2

1

4

Number of external nodes: at least h+1 and at most 2h

Number of internal nodes: at least h and at most 2h – 1 Total no. nodes: at least 2h + 1 and at most 2h+1 – 1 Height: at least log(n+1) – 1 and at most (n – 1)/2 Number of external nodes = number of internal nodes + 1

Height

1

2

0 n = 7

65

Preorder and Postorder Traversals of a Binary Tree

Special cases of the traversals of the general tree

General Tree

Algorithm preorder(T,v) visit v for each child of v do preorder(T,child)

Binary Tree

Algorithm binaryPreorder(T,v) visit v if v is an internal node then binaryPreorder(T,T.left(v)) binaryPreorder(T,T.right(v))

Algorithm postorder(T,v) for each child of v do postorder(T,child) visit v

Algorithm binaryPostorder(T,v) if v is an internal node then binaryPostorder(T,T.left(v)) binaryPostorder(T,T.right(v)) visit v


66

Inorder Traversal of a Binary TreeTrees (Goodrich, 279)

Algorithm inorder(T,v) if v is an internal node then inorder(T,T.left(v)) visit v if v is an internal node then inorder(T,T.right(v))

*

3

+

41 2

+

T

v

1

67



*

3

+

41 2

+

T

v

v

1+

68



*

3

+

41 2

+

T

v

v v

1+2

69



*

3

+

41 2

+

T

v

v

v v

1+2*

70



*

3

+

41 2

+

T

v

v

v v v

1+2*3

71



*

3

+

41 2

+

T

v

v

v v v

v

1+2*3+

72



*

3

+

41 2

+

T

v

v

v v v v

v

1+2*3+4

73

Euler Tour Traversal

Traversal of a binary tree that starts at the root and walks around the tree

Visits each node three times (left, below, and right)


vv

v

vv

v

vv

v

vv

v

vv

v vv

v

vv

v vv

v

vv

v

2

5 1

3 2

74

Implementing a Binary Tree

75

Linked Binary Tree

Natural way to represent a tree is by using linked nodes


76

Node for a Binary Tree

leftelement

parentright


77


leftelement

parentright

leftr

nullright

nulls

parentnull null

t

parentnull

root


78


public class BTNode<E> implements BTPosition { private E element; public BTNode() { } public BTNode(E element, BTPosition<E> parent,

BTPosition<E> left, BTPosition<E> right) { setElement(element); setParent(parent); setLeft(left); setRight(right); } public E element() { return element; } public void setElement(E o) { element=o; } public BTPosition<E> getLeft() { return left; } public void setLeft(BTPosition<E> v) { left=v; } public BTPosition<E> getRight() { return right; } public void setRight(BTPosition<E> v) { right=v; } public BTPosition<E> getParent() { return parent; } public void setParent(BTPosition<E> v) { parent=v; }}


79

Position for a Binary Tree

public interface BTPosition<E> extends Position<E>{// inherits element()

public void setElement(E o); public BTPosition<E> getLeft(); public void setLeft(BTPosition<E> v); public BTPosition<E> getRight(); public void setRight(BTPosition<E> v); public BTPosition<E> getParent(); public void setParent(BTPosition<E> v);}


80

A Position Interface

public interface Position<E> {

/** Return the element stored at this position. */

E element();

}

A Position object, has only one operation that it can use. This operation returns the element stored inside the object.

Positions (Goodrich, 232, 234)

81

Some Operations of a Binary Tree

public class LinkedBinaryTree<E> implements BinaryTree<E> { public int size() {return size; } public boolean isEmpty() { /*...*/ } public boolean isInternal(Position<E> v) throws InvalidPositionException { checkPosition(v); return (hasLeft(v) || hasRight(v)); } public boolean isExternal(Position<E> v) throws InvalidPositionException { /*...*/ } public boolean isRoot(Position<E> v) throws InvalidPositionException { checkPosition(v); return (v == root()); } public boolean hasLeft(Position<E> v) throws InvalidPositionException { BTPosition<E> vv = checkPosition(v); return (vv.getLeft() != null); } public boolean hasRight(Position<E> v) throws InvalidPositionException { /*...*/ } public Position<E> root() throws EmptyTreeException { /*...*/ } public Position<E> left(Position<E> v) throws InvalidPositionException, BoundaryViolationException { BTPosition<E> vv = checkPosition(v); Position<E> leftPos = vv.getLeft(); if (leftPos == null) throw new BoundaryViolationException(); return leftPos;} public Position<E> right(Position<E> v) throws InvalidPositionException, BoundaryViolationException { /*...*/ }


82

Instance Variablesof a Binary Tree

public class LinkedBinaryTree<E> implements BinaryTree<E> { protected BTPosition<E> root; // reference to the root protected int size; // number of nodes

//...


83

Constructor of a Binary Tree

public class LinkedBinaryTree<E> implements BinaryTree<E> { protected BTPosition<E> root; // reference to the root protected int size; // number of nodes

public LinkedBinaryTree() { root = null; // start with an empty tree size = 0; }

//...


84

Operations of the LinkedBinaryTree

Tell us about the positionspos root()pos parent( pos )iter children( pos )bool isInternal( pos )bool isExternal( pos )bool isRoot( pos )pos left( pos )pos right( pos )pos hasLeft( pos )pos hasRight( pos )pos sibling( pos )

Tell us about the collectionint size()bool isEmpty()iter iterator()iter positions()

Update the datavoid replace( pos, e )addRoot( e )insertLeft( pos, e )insertRight( pos, e )remove( pos )attach( pos, T1, T2 )


Tree methodsBinary Tree methodsAdditional methods

85

Adding an Element to a LinkedBinaryTree Object

LinkedBinaryTree<Character> T = new LinkedBinaryTree<Character>;

Creates an empty tree.


root

86





root

AT.addRoot(new Character('A')); Adds a node as root and fills it with an element.

root

87





root


root

AT.insertLeft( T.root(), new Character('B'));

Inserts the node with its element.

B

root

88

Creating a Tree with LinkedBinaryTree

A

F

C

GD E

B

Trees

89

Instantiating an EmptyLinkedBinaryTree Object

//Default constructorpublic LinkedBinaryTree(){ root = null; size = 0;}




root

90

Adding a Root and Element to a LinkedBinaryTree Object

public Position<E> addRoot(E e) throws NonEmptyTreeException {

if(!isEmpty()) throw new NonEmptyTreeException(); size = 1; root = createNode(e,null,null,null); return root; }



root

91

Adding Another Element to a LinkedBinaryTree Object

public Position<E> insertLeft(Position<E> v, E e) throws InvalidPositionException { BTPosition<E> vv = checkPosition(v); Position<E> leftPos = vv.getLeft(); if (leftPos != null) throw new InvalidPositionException(); BTPosition<E> ww = createNode(e, vv, null, null); vv.setLeft(ww); size++; return ww; }


AT.insertLeft( T.root(), new Character('B'));

Inserts the node with its element.

B

root

92

An Alternative Binary Tree Implementation

ArrayList Implementation

93

Binary TreeImplemented with an ArrayList

While it’s natural to think of implementing a tree with linked Nodes, a binary tree can also be implemented with an ArrayList

Advantage: better performance


94

Binary TreeImplemented as an ArrayList

Based on a way of numbering the nodes• If v is the root of T, then p(v) = 1• If v is the left child of node u, then p(v) = 2 * p(u)• If v is the right child of node u, then p(v) = 2 * p(u) + 1


D

E

F

GA C

B

1

2 3

4 5 6 7

D B F A C E G

0 1 2 3 4 5 6 7 8 9A

Example:

How would we implement root()?

95

References

Dale, N., C++ Plus Data Structures. Boston: Jones and Bartlett Publishers, 2003.

Goodrich, M. T. and R. Tamassia, Data Structures and Algorithms in Java. Hoboken, NJ: John Wiley & Sons, Inc., 2006.

Documents

1 Today’s Objectives Announcements Hand in Homework 3 next week Iterators Trees Tree terminology Depth and height Preorder and postorder traversals