cs225 ch4 trees and search trees -...

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1. Tree ADT

2. Binary Trees

3. Binary Search Trees3 y

4. AVL Trees

Semester I (2014) 1

What is a Tree

In computer science, a tree is an abstract model

Computers”R”Us

1. Tree ADT

of a hierarchical structure

A tree consists of nodes with a parent‐child relation

Applications:

Organization charts

File systems

Sales R&DManufacturing

Laptops DesktopsUS International

Semester I (2014) 2

File systems

Programming environments

Europe Asia Canada

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Tree Terminology Root: node without parent (A)

Internal node: node with at least one child (A, B, C, F)

Subtree: tree consisting of a node and its descendants

1. Tree ADT

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 d ( )

A

B DC

Semester I (2014) 3

any node (3)

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

G HE F

I J K

sub‐tree

We use positions to abstract nodes

h d

Query methods: boolean isInternal(p)

b l i E l( )

1. Tree ADT

Generic methods: integer size()

boolean isEmpty()

objectIterator elements()

positionIterator positions()

Accessor methods:

boolean isExternal(p)

boolean isRoot(p)

Update methods: swapElements(p, q)

object replaceElement(p, o)

Additional update h d b d fi d

Semester I (2014) 4

position root()

position parent(p)

positionIterator children(p)

methods may be defined by data structures implementing the Tree ADT

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Pre‐order Traversal

A traversal visits the nodes of a tree in a systematic manner

Algorithm preOrder(v)visit(v)

1. Tree ADT

In a preorder traversal, a node is visited before its descendants

Application: print a structured document

Make Money Fast!1

9

for each child w of vpreorder (w)

Semester I (2014) 5

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme

1.1 Greed 1.2 Avidity2.3 BankRobbery

2

3

5

4 6 7 8

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Pre‐order Traversal

Example: Preorder traversal of an ordered tree, where the children of each node are ordered from left to right.

1. Tree ADT

Semester I (2014) 6

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Post‐order Traversal

In a postorder traversal, a node is visited after its descendants

Algorithm postOrder(v)for each child w of v

1. Tree ADT

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

postOrder (w)visit(v)

cs16/9

3 78

Semester I (2014) 7

homeworks/todo.txt1K

programs/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

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1

7

2 4 5 6

Post‐order Traversal

Example: Postorder traversal

1. Tree ADT

Semester I (2014) 8

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Class Assignment Give a tree as the figure

below:

Answer the questions1. What is Root?

Computers”R”Us

Sales R&DManufacturing

2. What are internal nodes?

3. What are external nodes?

4. What are ancestors of the node US?

5. What is the depth of the node Asia?

6. What is height of the tree

7. What descendant of the

Laptops DesktopsUS International

Europe Asia Canada

7

node Sales?

8. What is pre‐order traversal of the tree?

9. What is post‐order traversal of the tree?

Semester I (2014) 9

A binary tree is a tree with the following properties: Each internal node has two children

The children of a node are an

Applications: arithmetic

expressions

decision processes

searching

2. 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

i i f i l d

searching

A

B C

Semester I (2014) 10

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

F GD E

H I

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Arithmetic Expression Tree

Binary tree associated with an arithmetic expression internal nodes: operators

2. Binary Trees

internal nodes: operators

external nodes: operands

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


2

a 1

3 b

Decision Tree

Binary tree associated with a decision process internal nodes: questions with yes/no answer

2. Binary Trees

internal nodes: questions with yes/no answer

external nodes: decisions

Example: dining decision

Want a fast meal?

Yes No


How about coffee? On expense account?

Starbucks Spike’s Al Forno Café Paragon

Yes No Yes No

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Properties of Binary Trees

Notation

n number of nodes

Properties:

e i 1

2. Binary Trees

e number of external nodes

i number of internal nodes

h height

n 2e 1

h i

h (n 1)2 e 2h

h log2 e


h log2 (n 1) 1

BinaryTree ADT

The BinaryTree ADT extends the Tree ADT

Update methods may b d fi d b d t

2. Binary Trees

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

Additional methods:

position leftChild(p)

position rightChild(p)

be defined by data structures implementing the BinaryTree ADT


position rightChild(p)

position sibling(p)

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Data Structure for Trees A node is represented by an object storing Element

1. Tree ADT

Element

Parent node

Sequence of children nodes

Node objects implement the Position ADT

B

B

A D F


DA

C E

F

C

E

Data Structure for Binary Trees A node is represented by

an object storing

Element

2. Binary Trees

Parent node

Left child node

Right child node

Node objects implement the Position ADT

B

B

A D


DA

C E

C E

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Auxiliary Structure Node

typedef int Object;struct Node {

Data Structure for Binary Trees

2. Binary Trees

struct Node {Object element;Node* parent;Node* left;Node* right;Node() : element(Object())

{ parent = left = right = NULL; }Node* sibling() const {return (this == parent ‐> left ? parent‐>right : parent‐> left);

}};typedef Node* NodePtr;


Position Class

class Position {

Data Structure for Binary Trees

2. Binary Trees

private:NodePtr node;

public:Position(NodePtr n = NULL){ node = n;}

Object& element() const{ t d l t }{ return node‐>element;}

bool isNull() const{return node == NULL;}

friend LinkedBinaryTree;};


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C++ Implementation

Tree interface

BinaryTree interface expandExternal(v)

2. Binary Trees

yextending Tree

Classes implementing Treeand BinaryTree and providing

Constructors

Update methods

Print methods

A

removeAboveExternal(w)

A

v v


Examples of updates for binary trees

expandExternal(v)

removeAboveExternal(w)

A

CB

Bw

The complete structure for LinkedBinaryTree

typedef int Elem; class LinkedBinaryTree { protected:// insert Node declaration here. . .public:

Position removeAboveExternal(const Position& p); // remove p and parent// housekeeping functions omitted. . .

protected: // local utilities

2. Binary Tree ADT: C++ Implementation

public:// insert Position declaration here. . .public:LinkedBinaryTree(); // constructorint size() const; // number of nodesbool empty() const; // is tree empty?Position root() const; // get the rootPositionList positions() const; // list of nodes

protected: // local utilitiesvoid preorder(Node* v, PositionList& pl) const; // preorder utilityprivate:Node* root; // pointer to the rootint n; // number of nodes};//… page 291 Textbook

//void addRoot(); // add root to empty treevoid expandExternal(const Position& p); // expand external node


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Class Assignment1. Write a C++ function that print post‐order

traversal of a binary tree to output screen.

2. Write a C++ function copy(T1 ,T2, v1, v2) that copy the node v2 of the tree T2 to the position of the node v1 of the tree T1.


3. Binary Search Tree

A binary search tree is a binary tree storing keys (or key‐element pairs) at its internal nodes and

An inorder traversal of a binary search trees visits the keys in increasing dits internal nodes and

satisfying the following property: Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have

order

6

92v


key(u) key(v) key(w)

External nodes do not store items

41 8u w

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Search To search for a key k, we

trace a downward path starting at the root

Algorithm find (k, v)if T.isExternal (v)

return Position(null)

3. Binary Search Trees

g

The next node visited depends on the outcome of the comparison of kwith the key of the current node

If we reach a leaf, the key is not found and we

if k key(v)return find (k, T.leftChild(v))

else if k key(v)return Position(v)

else { k key(v) }return find (k, T.rightChild(v))

6


return a null position

Example: find(4)

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Insertion To perform operation

insertItem(k, o), we search

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92


( , ),for key k

Assume k is not already in the tree, and let let w be the leaf reached by the search

We insert k at node w and expand w into an internal node

6

92

41 8

w


Example: insert 592

41 8

5w

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Deletion

To perform operation removeElement(k), we search

6

92


( ),for key k

Assume key k is in the tree, and let let v be the node storing k

If node v has a leaf child w, we remove v and w from the tree with operation

92

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5

vw

6


premoveAboveExternal(w)

Example: remove 492

51 8

Deletion (cont.)

We consider the case where the key k to be removed is stored at

3

1v


key k to be removed is stored at a node v whose children are both internal

we find the internal node w that follows v in an inorder traversal

we copy key(w) into node v

we remove node w and its left

8

6 9

5w

z

2

1


child z (which must be a leaf) by means of operation removeAboveExternal(z)

Example: remove 3

5

8

6 9

v

2

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Performance

Consider a dictionary with n items


with n items implemented by means of a binary search tree of height h the space used is O(n) methods find(), insertItem() and removeElement() take


O(h) time

The height h is O(n) in the worst case and O(log n) in the best case

Binary Search in Ordered Dictionary

Binary search performs operation find(k) on a dictionary implemented by means of an array‐based sequence, sorted by key


similar to the high‐low game

at each step, the number of candidate items is halved

terminates after O(log n) steps

Example: find(7)

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

0

0

ml h


1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

0

0

0

ml h

ml h

lm h

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Class Assignment

1. Insert into an initially empty binary search tree,


1. Insert into an initially empty binary search tree, items with the following key (in this order): 30, 40, 24, 58, 48, 26, 11, 13. Draw the tree after each insertion.


AVL Tree Definition

AVL trees are 4

4. AVL Tree

t ees a ebalanced.

An AVL Tree is a binary search treesuch that for every internal node v of T,

88

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17 78

32 50

48 62

2

4

1

1

2

3

1

1


the heights of the children of v can differ by at most 1. An example of an AVL tree where the

heights are shown next to the nodes:

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Height of an AVL Tree

Fact: The height of an AVL tree storing n keys is O(log n).

Proof: Let us bound n(h): the minimum number of internal

3

4 n(1)

n(2)4. AVL Tree

Proof: Let us bound n(h): the minimum number of internal nodes of an AVL tree of height h.

We easily see that n(1) = 1 and n(2) = 2

For n > 2, an AVL tree of height h contains the root node, one AVL subtree of height n‐1 and another of height n‐2.

That is, n(h) = 1 + n(h‐1) + n(h‐2)

Knowing n(h‐1) > n(h‐2), we get n(h) > 2n(h‐2). So


g ( ) ( ) g ( ) ( )n(h) > 2n(h‐2), n(h) > 4n(h‐4), n(h) > 8n(n‐6), … (by induction),

n(h) > 2in(h‐2i)

Solving the base case we get: n(h) > 2 h/2‐1

Taking logarithms: h < 2log n(h) +2

Thus the height of an AVL tree is O(log n)

Insertion in an AVL Tree

Insertion is as in a binary search tree Always done by expanding an external node.

4. AVL Tree

Example:44

17 78

32 50 88

48 62

a=y

c=z

44

17 78

32 50 88

48 62


48 62

54w

b=x48 62

before insertion after insertion

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Trinode Restructuring

let (a,b,c) be an inorder listing of x, y, z

perform the rotations needed to make b the topmost node of the three

4. AVL Tree

three

b=y

a=z

c=x

T0

T1 b=y

c=y

b=x

a=z

T0

T3b=x

case 2: double rotation(a right rotation about c, then a left rotation about a)

(other two cases are symmetrical)


T1

T2 T3

b=y

a=z c=x

T0 T1 T2 T3

T1 T2

b=x

c=ya=z

T0 T1 T2 T3

case 1: single rotation(a left rotation about a)

Insertion Example, continued44

17 782

5

1 3

4

1y

z

2

67

4. AVL Tree

8832 50

48 621 2

54

1

T0T2

T3

x3

4

5

1

44

17 622

4

3 x

y z

unbalanced...

2

4

6

T1


88

7832 50

48

1

1

2 2

1

54

1

T0 T1

T2

T3

y z

...balanced

1

2

3 5 7

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Restructuring (as Single Rotations)

Single Rotations:

4. AVL Tree

T0T1

T2

T3

c = xb = y

a = z

T0 T1 T2

T3

c = xb = y

a = zsingle rotation


T3T2

T1

T0

a = xb = y

c = z

T0T1T2

T3

a = xb = y

c = zsingle rotation

T0T1 T2 T3

Restructuring (as Double Rotations) double rotations:

4. AVL Tree

double rotationa = z

b = xc = y

T0T2

T1

T3 T0

T2T3T1

a = zb = x

c = y


double rotationc = z

b = xa = y

T0T2

T1

T3 T0

T2T3 T1

c = zb = x

a = y

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Removal in an AVL Tree

Removal begins as in a binary search tree, which means the node removed will become an empty external node. Its parent, w may cause an imbalance

4. AVL Tree

w, may cause an imbalance. Example:

44

17

7832 50

62

44

17

7850

62


8848 54 8848 54

before deletion of 32 after deletion

Rebalancing after a Removal

Let z be the first unbalanced node encountered while travelling up the tree from w. Also, let y be the child of z with the larger height,

4. AVL Tree

and let x be the child of y with the larger height.

We perform restructure(x) to restore balance at z.

As this restructuring may upset the balance of another node higher in the tree, we must continue checking for balance until the root of T is reached

44

17 62 b

a=z

44 78

62


17

7850

8848

62

54

w

c=x

b=y

17 50 88

48 54

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Running Times for AVL Trees a single restructure is O(1)

using a linked‐structure binary tree

4. AVL Tree

find is O(log n) height of tree is O(log n), no restructures needed

insert is O(log n) initial find is O(log n)

Restructuring up the tree, maintaining heights is O(log n)

remove is O(log n)


( g ) initial find is O(log n)

Restructuring up the tree, maintaining heights is O(log n)

Class Assignment

Balance the binary search tree below

4. AVL Tree

Balance the binary search tree below

5

1

8

6 9

2

10


10

12

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Class Assignment Balance the binary search tree below

14

5

8

6 9

2

10

121

2

3

4

5

15

8

6

9

2

10

121

2

3

4

1


cs225 ch4 trees and search trees -...

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