1 jake’s pizza shop owner jake manager chef brad carol waitress waiter cook helper joyce chris max...

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Jake’s Pizza Shop

Owner Jake

Manager Chef Brad Carol

Waitress Waiter Cook Helper Joyce Chris Max Len

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Owner Jake



A Tree Has a Root Node

ROOT NODE

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Owner Jake



Leaf nodes have no children

LEAF NODES

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Owner Jake



A Tree Has Levels

LEVEL 0

5

Owner Jake



Level One

LEVEL 1

6

Owner Jake



Level Two

LEVEL 2

7

Owner Jake



A Subtree

LEFT SUBTREE OF ROOT NODE

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Owner Jake



Another Subtree

RIGHT SUBTREE OF ROOT NODE

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A binary tree is a structure in which:

Each node can have at most two children, and in which a unique path exists from the root to every other node.

The two children of a node are called the left child and the right child, if they exist.

Binary Tree

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A Binary Tree

Q

V

T

K S

A E

L

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How many leaf nodes?

Q

V

T

K S

A E

L

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How many descendants of Q?

Q

V

T

K S

A E

L

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How many ancestors of K?

Q

V

T

K S

A E

L

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Implementing a Binary Tree with Pointers and Dynamic Data

Q

V

T

K S

A E

L

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Each node contains two pointers

struct TreeNode {

ItemType info; // Data member

TreeNode* left; // Pointer to left child

TreeNode* right; // Pointer to right child

};

. left . info . right

NULL ‘A’ 6000

// BINARY SEARCH TREE SPECIFICATION

class TreeType { public:

TreeType ( ); // constructor ~TreeType ( ); // destructor bool IsEmpty ( ) const; bool IsFull ( ) const; int NumberOfNodes ( ) const; void InsertItem ( ItemType item ); void DeleteItem (ItemType item ); void RetrieveItem ( ItemType& item, bool& found ); void PrintTree (ofstream& outFile) const;

. . .

private:

TreeNode* root;}; 16

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TreeType CharBST;

‘J’

‘E’

‘A’

‘S’

‘H’

TreeType

~TreeType

IsEmpty

InsertItem

Private data:

root

RetrieveItem

PrintTree . . .

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A special kind of binary tree in which:

1. Each node contains a distinct data value,

2. The key values in the tree can be compared using “greater than” and “less than”, and

3. The key value of each node in the tree is

less than every key value in its right subtree, and greater than every key value in its left subtree.

A Binary Search Tree (BST) is . . .

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Depends on its key values and their order of insertion.

Insert the elements ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ in that order.

The first value to be inserted is put into the root node.

Shape of a binary search tree . . .

‘J’

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Thereafter, each value to be inserted begins by comparing itself to the value in the root node, moving left it is less, or moving right if it is greater. This continues at each level until it can be inserted as a new leaf.

Inserting ‘E’ into the BST

‘J’

‘E’

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Begin by comparing ‘F’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

Inserting ‘F’ into the BST

‘J’

‘E’

‘F’

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Begin by comparing ‘T’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

Inserting ‘T’ into the BST

‘J’

‘E’

‘F’

‘T’

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Begin by comparing ‘A’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

Inserting ‘A’ into the BST

‘J’

‘E’

‘F’

‘T’

‘A’

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is obtained by inserting

the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order?

What binary search tree . . .

‘A’

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obtained by inserting

the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order.

Binary search tree . . .

‘A’

‘E’

‘F’

‘J’

‘T’

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Another binary search tree

Add nodes containing these values in this order:

‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’

‘K’ ‘P’

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Is ‘F’ in the binary search tree?

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’

‘K’

‘V’

‘P’ ‘Z’‘D’

‘Q’‘L’‘B’

‘S’

// BINARY SEARCH TREE SPECIFICATION

class TreeType { public:

TreeType ( ) ; // constructor ~TreeType ( ) ; // destructor bool IsEmpty ( ) const ; bool IsFull ( ) const ; int NumberOfNodes ( ) const ; void InsertItem ( ItemType item ) ; void DeleteItem (ItemType item ) ; void RetrieveItem ( ItemType& item , bool& found ) ; void PrintTree (ofstream& outFile) const ;

. . .

private:

TreeNode* root ;}; 28

// SPECIFICATION (continued) // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -// RECURSIVE PARTNERS OF MEMBER FUNCTIONS

void PrintHelper ( TreeNode* ptr, ofstream& outFile ) ;

void InsertHelper ( TreeNode*& ptr,

ItemType item ) ;

void RetrieveHelper ( TreeNode* ptr,

ItemType& item, bool& found ) ;

void DestroyHelper ( TreeNode* ptr ) ;

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// BINARY SEARCH TREE IMPLEMENTATION // OF MEMBER FUNCTIONS AND THEIR HELPER FUNCTIONS

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -TreeType :: TreeType ( ) // constructor { root = NULL ;}

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -bool TreeType :: IsEmpty( ) const{ return ( root == NULL ) ;}

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void TreeType :: RetrieveItem ( ItemType& item, bool& found )

{ RetrieveHelper ( root, item, found ) ; }

void RetrieveHelper ( TreeNode* ptr, ItemType& item, bool& found)

{ if ( ptr == NULL ) found = false ;

else if ( item < ptr->info ) // GO LEFTRetrieveHelper( ptr->left , item, found ) ;

else if ( item > ptr->info ) // GO RIGHTRetrieveHelper( ptr->right , item, found ) ;

else { item = ptr->info ;

found = true ; }} 31

void TreeType :: InsertItem ( ItemType item ){

InsertHelper ( root, item ) ; }

void InsertHelper ( TreeNode*& ptr, ItemType item ){ if ( ptr == NULL ) { // INSERT item HERE AS LEAF

ptr = new TreeNode ;ptr->right = NULL ;ptr->left = NULL ;ptr->info = item ;

} else if ( item < ptr->info ) // GO LEFT

InsertHelper( ptr->left , item ) ; else if ( item > ptr->info ) // GO RIGHT

InsertHelper( ptr->right , item ) ;}

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Inorder Traversal: A E H J M T Y

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print left subtree first Print right subtree last

Print second

// INORDER TRAVERSAL

void TreeType :: PrintTree ( ofstream& outFile ) const{

PrintHelper ( root, outFile ) ; }

void PrintHelper ( TreeNode* ptr, ofstream& outFile ){ if ( ptr != NULL ) {

PrintHelper( ptr->left , outFile ) ; // Print left subtree

outFile << ptr->info ;

PrintHelper( ptr->right, outFile ) ; // Print right subtree

}} 34

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Preorder Traversal: J E A H T M Y

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print left subtree second Print right subtree last

Print first

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‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’ ‘Y’

tree

Print left subtree first Print right subtree second

Print last

Postorder Traversal: A H E M Y T J

TreeType :: ~TreeType ( ) // DESTRUCTOR{ DestroyHelper ( root ) ; }

void DestroyHelper ( TreeNode* ptr )// Post: All nodes of the tree pointed to by ptr are deallocated.

{ if ( ptr != NULL ) {

DestroyHelper ( ptr->left ) ;DestroyHelper ( ptr->right ) ;delete ptr ;

}}

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Deleting a Node

Find node (return parent and node) Four Cases: Case 1 leaf node Case 2 no left subtree Case 3 no right subtree Case 4 left and right subtrees

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Case 1 Leaf Node

Point parent to NULL Delete node

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Case 2 No Left Subtree

Point parent to node right subtree Delete node

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Case 3 No Right Subtree

Point parent to left subtree Delete node

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Case 4 Left and Right Subtrees

Find node with greatest key in left subtree (or smallest key in right subtree)

Copy greatest (or smallest) key node data to delete node data

Delete greatest (or smallest) key node

1 jake’s pizza shop owner jake manager chef brad carol waitress waiter cook helper joyce chris max...

Documents

binary tree slide

root node root node

j slide

bst j e slide

children leaf nodes

binary tree q v t

binary search tree bst

right child