trees1 make money fast! stock fraud ponzi scheme bank robbery © 2010 goodrich, tamassia

Trees 1

Trees

Make Money Fast!

StockFraud

PonziScheme

BankRobbery

© 2010 Goodrich, Tamassia

Trees 2

What is a Tree In computer science,

a tree is an abstract model of a hierarchical structure

A tree consists of nodes with a parent-child relation

Applications: Organization charts File systems Programming

environments

Computers”R”Us

Sales R&DManufacturing

Laptops DesktopsUS International

Europe Asia Canada


Trees 3subtree

Tree Terminology Root: node without parent (A) Internal node: node with at least

one child (A, B, C, F) External node (a.k.a. leaf ):

node without children (E, I, J, K, G, H, D)

Ancestors of a node: parent, grandparent, grand-grandparent, etc.

Depth of a node: number of ancestors

Height of a tree: maximum depth of an external node (3)

Descendant of a node: child, grandchild, grand-grandchild, etc.

A

B DC

G HE F

I J K

Subtree: tree consisting of a node and its descendants


4

More TerminologyTrees (Goodrich, 268)

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

Subtree rooted at v

vv

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

Trees 5

Tree ADT We use positions to

abstract nodes Generic methods:

integer size() boolean isEmpty() Iterator iterator() Iterable positions()

Accessor methods: position root() position parent(p) Iterable children(p)

Query methods: boolean isInternal(p) boolean isExternal(p) boolean isRoot(p)

Update method: element replace (p, o)

Additional update methods may be defined by data structures implementing the Tree ADT


6

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;

}

Trees (Goodrich, 270)

7

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

8

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

9

Finding the Depth

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


10

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) );}

11

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


12

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


13

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


14



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


15



u

t


r 0 0

1

2


t

return 0

return 0u

T


16



u

t


r 0 0

1

2


t

return 0 return 0

return 0ureturn 0v

T


17



u

t


r 0 0

1

2


t

return 0

return 0u

T


return 0

return 0v

return 1

18



u

t


r 0 0

1

2

return 1

return 2


t

return 0 return 0

return 0ureturn 0v

T


Trees 19

Preorder Traversal A traversal visits the nodes of

a tree in a systematic manner In a preorder traversal, a

node is visited before its descendants

Application: print a structured document

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1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme

1.1 Greed 1.2 Avidity2.3 BankRobbery

1

2

3

5

4 6 7 8

9

Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)


Trees 20

Postorder Traversal In a postorder traversal, a

node is visited after its descendants

Application: compute space used by files in a directory and its subdirectories

Algorithm postOrder(v)for each child w of v

postOrder (w)visit(v)

cs16/

homeworks/todo.txt

1Kprograms/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

9

3

1

7

2 4 5 6

8


Trees 21

Binary Trees A binary tree is a tree with the

following properties: Each internal node has at most

two children (exactly two for proper binary trees)

The children of a node are an ordered pair

We call the children of an internal node left child and right child

Alternative recursive definition: a binary tree is either

a tree consisting of a single node, or

a tree whose root has an ordered pair of children, each of which is a binary tree

Applications: arithmetic

expressions decision processes searching

A

B C

F GD E

H I© 2010 Goodrich, Tamassia

Trees 22

Arithmetic Expression Tree Binary tree associated with an arithmetic expression

internal nodes: operators external nodes: operands

Example: arithmetic expression tree for the expression (2 (a 1) (3 b))

2

a 1

3 b


Trees 23

Decision Tree Binary tree associated with a decision process

internal nodes: questions with yes/no answer external nodes: decisions

Example: dining decision

Want a fast meal?

How about coffee? On expense account?

Starbucks Spike’s Al Forno Café Paragon

Yes No

Yes No Yes No


Trees 24

Properties of Proper Binary Trees

Notationn number of nodese number of

external nodesi number of

internal nodesh height

Properties: e i 1 n 2e 1 h i h (n 1)2 e 2h

h log2 e

h log2 (n 1) 1


Level d has at most 2d nodes

© 2010 Goodrich, Tamassia Trees 25

Trees 26

BinaryTree ADT

The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT

Additional methods: position left(p) position right(p) boolean hasLeft(p) boolean hasRight(p)

Update methods may be defined by data structures implementing the BinaryTree ADT


Trees 27

Inorder Traversal In an inorder traversal a

node is visited after its left subtree and before its right subtree

Application: draw a binary tree

x(v) = inorder rank of v y(v) = depth of v

Algorithm inOrder(v)if hasLeft (v)

inOrder (left (v))visit(v)if hasRight (v)

inOrder (right (v))

3

1

2

5

6

7 9

8

4


Trees 28

Print Arithmetic Expressions

Specialization of an inorder traversal

print operand or operator when visiting node

print “(“ before traversing left subtree

print “)“ after traversing right subtree

Algorithm printExpression(v)if hasLeft (v)

print(“(’’)inOrder (left(v))

print(v.element ())if hasRight (v)

inOrder (right(v))print (“)’’)

2

a 1

3 b((2 (a 1)) (3 b))


Trees 29

Evaluate Arithmetic Expressions

Specialization of a postorder traversal

recursive method returning the value of a subtree

when visiting an internal node, combine the values of the subtrees

Algorithm evalExpr(v)if isExternal (v)

return v.element ()else

x evalExpr(leftChild (v))

y evalExpr(rightChild (v))

operator stored at vreturn x y

2

5 1

3 2


Trees 30

Euler Tour Traversal Generic traversal of a binary tree Includes a special cases the preorder, postorder and inorder

traversals Walk around the tree and visit each node three times:

on the left (preorder) from below (inorder) on the right (postorder)

2

5 1

3 2

LB

R


Trees 31

Linked Structure for Trees A node is represented

by an object storing Element Parent node Sequence of children

nodes Node objects implement

the Position ADT

B

DA

C E

F

B

A D F

C

E© 2010 Goodrich, Tamassia

Trees 32

Linked Structure for Binary Trees A node is represented

by an object storing Element Parent node Left child node Right child node

Node objects implement the Position ADT

B

DA

C E

B

A D

C E


Array-Based Representation of Binary Trees

Nodes are stored in an array A

© 2010 Goodrich, Tamassia 33Trees

Node v is stored at A[rank(v)] rank(root) = 1 if node is the left child of parent(node),

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

if node is the right child of parent(node), rank(node) = 2

rank(parent(node)) 1

1

2 3

6 74 5

10 11

A

HG

FE

D

C

B

J

A B D G H ……

1 2 3 10 110

34

Implementing a Binary Tree

35

Linked Binary Tree

Natural way to represent a tree is by using linked nodes


36

Node for a Binary Tree

leftelement

parentright


37


leftelement

parentright

leftr

nullright

nulls

parentnull null

t

parentnull

root


38


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; }}


39

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);}


40

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)

41

Some Operations of a Binary TreeReturn to pages 304 - 308

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 { /*...*/ }


42

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

//...


43

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; }

//...


44

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 )


Tree methodsBinary Tree methodsAdditional methods

45

Adding an Element to a LinkedBinaryTree Object

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

Creates an empty tree.


root

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

root

46

Adding an Element to a LinkedBinaryTree Object

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

Creates an empty tree.


root

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

root

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

Inserts the node with its element.

B

root

47

An Alternative Binary Tree Implementation

ArrayList Implementation

48

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


49

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

trees1 make money fast! stock fraud ponzi scheme bank robbery © 2010 goodrich, tamassia

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