1 chapter 4 trees basic concept how tree are used to implement the file system how tree can be used...

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Chapter 4 Trees

• Basic concept• How tree are used to implement the file

system• How tree can be used to evaluate

arithmetic expressions• How to use trees to support searching

operations in O(logN) average time

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4.1: Preliminaries

One natural way to define a tree is recursively. A tree is a collection of nodes. The collection can be empty; otherwise, a tree consists of a distinguished node r, called root, and zero or more nonempty (sub)trees T1, T2, … Tk, each of whose roots are connected by a direct edge from r.

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4.1: Preliminaries

• ParentNode A is the parent of node B if B is the root of the left or right subtree of A

• Left (Right) ChildNode B is the left (right) child of node A if A is the parent of B.

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4.1: Preliminaries

• SiblingNode B and node C are siblings if they are left and right siblings of the same node A

• LeafA node is called a leaf if it has no children

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4.1: Preliminaries

• AncestorNode A is ancestor of node B if A is either the parent of B or is the parent of some ancestor of B

• Left (Right) DescendantNode B is the left (right) descendant of node A if B is either the left (right) child of A or a descendant of a left (right) child of A

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4.1: Preliminaries

• Level of a NodeLevel of the root of a binary tree is 0, and the level of any other node in the tree is one more than the level of its parent

• Depth of a TreeThe depth of a tree is the maximum level of any leaf in the tree (also called the height of the tree)

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4.1.1 Preliminaries: Implementation of Tree

typedef struct TreeNode * PtrToNode;struct TreeNode{ ElementType element;

PtrToNode FirstChild;PtrToNode NextSibling;

}

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Example

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FirstChild pointer: arrow that points downward

NextSibling pointer: arrow that goes left to right

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4.1.1 Preliminaries: Unix File System

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• Any suggestions to handle such a file system?

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Static void ListDir(DirectoryOrFile D, int Depth){ if (D is a legitimate entry)

{ PrintName(D, Depth);if (D is a directory)for each child,C, of DListDir(C, Depth +1);}

}void ListDirectory(DirectoryOrFile D){ListDir (D, 0); }

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Static void SizeDirectory(DirectoryOrFile D){ int TotalSize;

TotalSize = 0;if (D is a legitimate entry){ TotalSize = FileSize(D);if (D is a directory)for each child, C, of DTotalSize += SizeDirectory(C); }

return TotalSize; }

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4.1.1 Preliminaries: Traversal Strategy

• 3 methods of traversal postorder traversal strategypreorder traversal strategyinorder traversal strategy

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Preorder traversal (depth-first order)1.Visit the node2.Traverse the left subtree in preorder3.Traverse the right subtree in preorder

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• Inorder traversal (symmetric order)1. Traverse the left subtree in inorder2. Visit the node3. Traverse the right subtree in inorder

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• Postorder traversal (breadth-first order)1. Traverse the left subtree in postorder2. Traverse the right subtree in

postorder3. Visit the node

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+

- H

/ *

+ * E -

A B C D F G

Example

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+

- H

/ *

+ * E -

A B C D F G

Preorder: +-/+AB*CD*E-FGHInorder : A+B/C*D-E*F-G+HPostorder: AB+CD*/EFG-*-H+

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• Exercise:• Using the tree shown in page 9, write

down the results if employing thepostorder traversal strategypreorder traversal strategyinorder traversal strategy

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4.2 Binary Trees

A binary tree is a finite set of elements called nodes. This set is either empty or it can be partitioned into 3 subsets. The first subset is a single node called the root of the tree. The other two subsets are themselves binary trees. One is called the left subtree and the other the right subtree of the binary tree.

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4.2 Binary Trees

• Strictly Binary TreeA binary tree is called strictly binary tree if every nonleaf node in the tree has nonempty left and right subtrees

• Complete (Full) Binary TreeA complete binary tree of depth d is a strictly binary tree with all leaf nodes at level d.

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4.2 Binary Trees

• Almost Complete Binary TreeA binary tree of depth d is an almost complete binary tree if(a) Any node at level less than d-1 has 2 children.(b) For any node N in the tree with a right descendant at level d, N must have a left child and every left descendant of N is either a leaf at level d or has 2 children.

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4.2 Binary Trees

An almost complete strictly binary tree with n leaves has 2n - 1 nodes.

There is only a single almost complete binary tree with n nodes. This tree is strictly binary if and only if n is odd.

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4.2 Binary Trees

• Strictly binary but not almost complete

Level 1, 2, 3

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4.2 Binary Trees

• Strictly binary but not almost complete

Violate condition 2

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4.2 Binary Trees

• Strictly binary and almost complete

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4.2 Binary Trees

• Almost complete but not strictly binary

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4.3 Binary Search Trees

• Every node, X, in the tree, the value of all the keys in its left subtree are smaller than the key value of X, and the values of all the keys in its right subtree are larger than the key value of X.

X

Please refer to figure 4.15

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4.3.1 BST: Implementation

struct TreeNode;typedef struct TreeNode *Position;typedef struct TreeNode *SearchTree;SearchTree MakeEmpty( SearchTree T );Position Find( ElementType X, SearchTree T );Position FindMin( SearchTree T );Position FindMax( SearchTree T );SearchTree Insert( ElementType X, SearchTree T );SearchTree Delete( ElementType X, SearchTree T );ElementType Retrieve( Position P );

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4.3.1 BST: Implementation

struct TreeNode{

ElementType Element;SearchTree Left;SearchTree Right;

};

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4.3.1 BST Implementation: MakeEmpty

SearchTree MakeEmpty( SearchTree T ) { if( T != NULL ) { MakeEmpty( TLeft ); MakeEmpty( T Right ); free( T ); } return NULL; }

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4.3.1 BST Implementation: Find

Position Find( ElementType X, SearchTree T ){ if ( T == NULL ) return NULL; if ( X < T Element ) return Find( X, T Left ); else if ( X > T Element ) return Find( X, T Right ); else return T; }

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4.3.1 BST Implementation: FindMin

Position FindMin( SearchTree T ) { if ( T == NULL ) return NULL; else if( T Left == NULL ) return T; else return FindMin( T Left );}

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4.3.1 BST Implementation: FindMax

Position FindMax( SearchTree T ){

if ( T != NULL ) while ( T Right != NULL ) T = T Right;

return T;}

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4.3.2 BST Implementation: Insert

• Please refer to Figure 4.21

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SearchTree Insert( ElementType X, SearchTree T ){ if ( T == NULL ) { T = malloc( sizeof( struct TreeNode ) ); if ( T == NULL )

FatalError( "Out of space!!!" ); else

{ T Element = X;

T Left = T Right = NULL; } }

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else

if ( X < T Element )

T Left = Insert( X, T Left ); else

if( X > T Element )

T Right = Insert( X, T Right ); /* Else X is in the tree already; we'll do nothing */

return T; /* Do not forget this line!! */ }

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4.3.3 BST Implementation: Delete

• 3 cases (please refer to Figure 4.24)

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4.3.3 BST Implementation: Delete SearchTree Delete( ElementType X, SearchTree T ) { Position TmpCell; if ( T == NULL )

Error( "Element not found" ); else

if ( X < T Element ) /* Go left */

T Left = Delete( X, T Left ); else

if ( X > T Element ) /* Go right */

T Right = Delete( X, T Right ); else /* Found element to be deleted */

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4.3.3 BST Implementation: Delete if ( T Left && T Right ) /* Two children */

{ TmpCell = FindMin( T Right );

T Element = TmpCell Element;

T Right = Delete( T Element, T Right );} else /* One or zero children */ { TmpCell = T;

if ( T Left == NULL ) /* Also handles 0 children */

T = T Right;

else if ( T Right == NULL ) T = T Left; free( TmpCell );} return T;}

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4.4 : Expression Tree

Expression tree for (a+b*c) +((d*e+f)*g)

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Algorithm to convert postfix expression into expression treee.g. input: ab+cde+**

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1 chapter 4 trees basic concept how tree are used to implement the file system how tree can be used...

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