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Chapter 7 - Tree

Basic tree concepts

Binary trees

Binary Search Tree (BST)

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Expression Trees

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Binary Tree ADT

DEFINITION: A binary tree ADT is either empty, or it consists

of a node called root together with two binary trees called

the left and the right subtree of the root.

Basic operations:

Constructa tree, leaving it empty.

Insertan element.

Remove an element.

Search an element.

Retrieve an element.

Traverse the tree, performing a given operation on each

element. 13

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Binary Tree ADT

Extended operations:

Determine whether the tree is emptyor not.

Find the size of the tree.

Clearthe tree to make it empty.

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Specifications for Binary Tree

Create()

isFull()

isEmpty()

Size()

Clear()

Search (ref DataOut )

Insert (val DataIn )

Remove (val key )

Retrieve (ref DataOut )

15

Depend on various

types of binary

trees

(BST, AVL, 2d-tree)

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Specifications for Binary Tree

Binary Tree Traversal: Each node is processed once and

only once in a predetermined sequence.

Depth-First Traverse:

preOrderTraverse (refOperation(ref Data ))

inOrderTraverse (refOperation(ref Data ))

postOrderTraverse (refOperation(ref Data ))

Breadth-First Traverse:

BreadthFirstTraverse (refOperation(ref Data ))16

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Contiguous Implementation of Binary Tree

0 1 2 3 4 5 6

A B C D E F G ...

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Physical Conceptual

(suitable for complete tree,

nearly complete tree, and

bushy tree)

2i+1 2i+2

i

BinaryTreeData

End BinaryTree

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Conceptual

Physical (suitable for sparse tree)

0 1 2 3 4 5 6

Data A B E C F G H ...

Parent - A B A C C F ...

Flag - L R R L R R ...

H

BinaryTree

Data

End BinaryTree

Record

Data

Parent

Flag

End Record

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Physical Conceptual

(A binary tree without identical data can be restored from

two array of LNR and NLR traverse)

0 1 2 3 4 5 6

NLR A B E C F H G

LNR B E A F H C G H

BinaryTree

NLR

LNR

End BinaryTree

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Linked Implementation of Binary Tree

BinaryNode

data

left

right

End BinaryNode

BinaryTree

root End BinaryTree

Physical 27

A

B

E

C

F G

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Depth-First Traversal

Auxiliary functions for Depth_First Traversal:

recursive_preOrder

recursive_inOrder

recursive_postOrder

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PreOrder Traversal

Algorithm recursive_preOrder (val subroot ,

refOperation(ref Data ))

Traverses a binary tree in node-left-rightsequence.

Pre subroot points to the root of a tree/ subtree.

Post each node has been processed in order.

1. if(subroot is not NULL)

1. Operation(subroot->data)

2. recursive_preOrder(subroot->left)

3. recursive_preOrder(subroot->right)

End recursive_preOrder29

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InOrder Traversal

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Algorithm recursive_inOrder (val subroot ,


Traverses a binary tree in left-node-rightsequence

Pre subroot points to the root of a tree/ subtree

Post each node has been processed in order


1. recursive_inOrder(subroot->left)


3. recursive_inOrder(subroot->right)

End recursive_inOrder30

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PostOrder Traversal

Algorithm recursive_postOrder (val subroot ,


Traverses a binary tree in left-right-node sequence

Pre subroot points to the root of a tree/ subtree



1. recursive_postOrder(subroot->left)

2. recursive_postOrder(subroot->right)


End recursive_postOrder31

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Depth-First Traversal preOrderTraverse (refOperation(ref Data ))

1. recursive_preOrder(root, Operation)

End preOrderTraverse

inOrderTraverse (refOperation(ref Data ))

1. recursive_inOrder(root, Operation)

End inOrderTraverse

postOrderTraverse (refOperation(ref Data ))

1. recursive_postOrder(root, Operation)

End postOrderTraverse 32

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Breadth-First Traversal

Algorithm BreadthFirstTraverse

(refOperation(ref Data ))

Traverses a binary tree in sequence from lowest level to highest

level, in each level traverses from left to right.


Uses Queue ADT

34

B dth Fi t T l

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Breadth-First TraversalAlgorithm BreadthFirstTraverse

(refOperation(ref Data ))1. if(root is not NULL)

1. queueObj

2. queueObj.EnQueue(root)

3. loop (not queueObj.isEmpty())

1. queueObj.QueueFront(pNode)

2. queueObj.DeQueue()

3. Operation(pNode->data)

4. if(pNode->left is not NULL)

1. queueObj.EnQueue(pNode->left)

5. if(pNode->right is not NULL)

1. queueObj.EnQueue(pNode->right)

End BreadthFirstTraverse 35

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Binary Search Tree

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Binary Search Tree

37

BST is one of implementations for ordered list.

In BST we can search quickly (as with binary search

on a contiguous list).

In BST we can make insertions and deletions quickly(as with a linked list).

When a BST is traversed in inorder, the keys will

come out in sorted order.

3


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Auxiliary functions for Search:

recursive_Search

iterative_Search

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( )

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Search node in BST (recursive version)

recursive_Search (val subroot ,

val target )Searches target in the subtree.


Post Iftarget is not in the subtree, NULL is returned. Otherwise,

a pointer to the node containing the target is returned.

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h d ( )

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Search node in BST (recursive version)1. if(subroot is NULL) OR (subroot->data = target)

1. return subroot2. else if(target < subroot->data)

1. return recursive_Search(subroot->left, target)

3. else

1. return recursive_Search(subroot->right, target)

End recursive_Search

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subroot

Target = 22

h d i ( i i )

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3. else



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subroot

Target = 22

S h d i S ( i i )

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3. else



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subroot

Target = 22

S h d i BST ( i i )

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3. else



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subroot

Target = 22

S h N d i BST ( i i )

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Search Node in BST (nonrecursive version)

iterative_Search (val subroot ,

val target )Searches target in the subtree.


Post Iftarget is not in the subtree, NULL is returned. Otherwise,

a pointer to the node containing the target is returned.

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Search Node in BST (nonrecursive version)1. while (subroot is not NULL) AND (subroot->data.key target)

1. if(target < subroot->data.key)1. subroot = subroot->left

2. else

1. subroot = subroot->right

2. return subroot

End iterative_Search

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subroot

Target = 22


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2. else


2. return subroot


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subroot

Target = 22


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2. else


2. return subroot


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subroot

Target = 22


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2. else


2. return subroot


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subroot

Target = 22

S h d i BST

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Search node in BST

Search (refDataOut )

Searches target in the subtree.

Pre DataOut contains value needs to be found in key field.

Post DataOut will reveive all other values in other fields if that

key is found.

Return success ornotPresent

UsesAuxiliary function recursive_Search oriterative_Search

1. pNode = recursive_Search(root, DataOut.key)

2. if(pNode is NULL)1. return notPresent

3. dataOut = pNode->data

4. return success

End Search 49

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S h d i BST

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Search node in BST

The same keys may be built into BST ofmany different

shapes.

Search in bushy BST with n nodes will do O(log n)

comparisons of keys

If the tree degenerates into a long chain, search will do (n)

comparisons on n vertices.

The bushier the tree, the smaller the number of comparisons of

keys need to be done.

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Insert Node into BST

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Insert Node into BST ?

?

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Question:

Can Insert method use recursive_Search or iterative_Search

instead ofrecursive_Insert like that:

53


1. pNode = recursive_Search (root, DataIn.key)

2. if(pNode is NULL)

1. Allocate pNode

2. pNode->data = DataIn

3. return success3. else

1. return duplicate_error

End Insert

??


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Auxiliary functions for Insert:

recursive_Insert

iterative_Insert

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Recursive Insert

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Recursive Insert

recursive_Insert (refsubroot ,

val DataIn )Inserts a new node into a BST.


DataIn contains data to be inserted into the subtree.

Post If the key ofDataIn already belongs to the subtree,

duplicate_erroris returned. Otherwise, DataIn is inserted

into the subtree in such a way that the properties of a BSTare preserved.

Return duplicate_errororsuccess.

Uses recursive_Insert function.

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Recursive Insert (cont )

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Recursive Insert (cont.) recursive_Insert (refsubroot ,

val DataIn )

1. if(subroot is NULL)

1. Allocate subroot

2. subroot->data = DataIn

3. return success

2. else if(DataIn.key < subroot->data.key)

1. return recursive_Insert(subroot->left, DataIn)

3. else if(DataIn.key > subroot->data.key)

1. return recursive_Insert(subroot->right, DataIn)

4. else


5. End recursive Insert 56

subroot

DataIn.key = 22


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val DataIn )


1. Allocate subroot


3. return success





4. else



subroot

DataIn.key = 22


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val DataIn )


1. Allocate subroot


3. return success





4. else



subroot

DataIn.key = 22

Recursive Insert (cont.)

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val DataIn )


1. Allocate subroot


3. return success





4. else



subroot

DataIn.key = 22

Recursive Insert (cont.)

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val DataIn )


1. Allocate subroot


3. return success





4. else



subroot

DataIn.key = 22

Iterative Insert

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Iterative Insert

iterative_Insert (refsubroot ,

val DataIn )

Inserts a new node into a BST.

Pre subroot is NULL or points to the root of a subtree. DataIn

contains data to be inserted into the subtree.

Post If the key ofDataIn already belongs to the subtree,

duplicate_erroris returned. Otherwise, DataIn is inserted into

the subtree in such a way that the properties of a BST arepreserved.

Return duplicate_erroror success.

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Iterative Insert (cont )

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Iterative Insert (cont.)

iterative_Insert (refsubroot ,

val DataIn )1. if(subroot is NULL)

1. Allocate subroot


3. return success

2. else

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subroot

DataIn.key = 22

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( )2. else

1. pCurr = subroot

2. loop (pCurr is not NULL)

1. if(pCurr->data.key = DataIn.key)


2. parent = pCurr

3. if(DataIn.key < parent->data.key)

1. pCurr = parent -> left4. else

1. pCurr = parent -> right


1. Allocate parent->left

2. parent->left.data = DataIn4. else

1. Allocate parent->right

2. parent->right.data = DataIn

5. return success

End Iterative_Insert 64

pCurr

DataIn.key = 22

subroot


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( )2. else

1. pCurr = subroot




2. parent = pCurr









5. return success


parent

pCurr

DataIn.key = 22

subroot


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( )2. else

1. pCurr = subroot




2. parent = pCurr









5. return success


parent

pCurr

DataIn.key = 22

subroot


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( )2. else

1. pCurr = subroot




2. parent = pCurr









5. return success


parent

pCurr

DataIn.key = 22

subroot


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( )2. else

1. pCurr = subroot




2. parent = pCurr









5. return success


parent

pCurr

DataIn.key = 22

subroot


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Inserts a new node into a BST.

Post If the key ofDataIn already belongs to the BST, duplicate_erroris

returned. Otherwise, DataIn is inserted into the tree in such a way

that the properties of a BST are preserved.

Return duplicate_erroror success.

Uses recursive_Insert or iterative_Insert function.

1. return recursive_Insert (root, DataIn)

End Insert

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Delete node from BST

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Deletion of a leaf:

Set the deleted node's parent link to NULL.

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Deletion of a node having only right subtree or left subtree:

Attach the subtree to the deleted node's parent.

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Deletion of a node having both subtrees:

Replace the deleted node by its predecessor or by its successor,

recycle this node instead.

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Replace X by W

W is predecessorof X

Delete

original W


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Node having both subtrees

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Node having both subtrees

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Auxiliary functions for Insert:

recursive_Delete

iterative_Delete

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Recursive Delete

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Recursive Delete

recursive_Delete (refsubroot ,

val key )Deletes a node from a BST.

Pre subroot is NULL or points to the root of a subtree. Key contains value

needs to be removed from BST.

Post Ifkey is found, it will be removed from BST.

Return notFoundor success.

Uses recursive_Delete and RemoveNode functions.

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Delete Node from BST

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Delete Node from BST

Delete (val key )

Deletes a node from a BST.

Pre subroot is NULL or points to the root of a subtree. Key contains value

needs to be removed from BST.

Post Ifkey is found, it will be removed from BST.Return notFoundor success.

Uses recursive_Delete and RemoveNode functions.

1. return recursive_Delete (root, key)

End Delete

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Auxiliary Function RemoveNode

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RemoveNode (refsubroot , val key )

1. pDel = subroot // remember node to delete at end.

2. if(subroot ->left is NULL) // leaf node or node having only right subtree.

1. subroot = subroot->right // (a) and (b)

3. else if(subroot->right is NULL) // node having only left subtree.

1. subroot = subroot->left

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subroot

pDel

key needs to be deleted = 18

subroot

pDel

(a)

(b)


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1. subroot = subroot->right // (a) and (b)


1. subroot = subroot->left

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subroot

pDel

key needs to be deleted = 18key needs to be deleted = 18

subroot

pDel

(a)

(b)


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1. subroot = subroot->left // (c)

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subrootpDel

key needs to be deleted = 44 (c)

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Auxiliary Function RemoveNode (cont.)

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4. else // node having both subtrees. (d)

1. parent = subroot

2. pDel = parent ->left // move left to find the predecessor.

3. loop (pDel->right is not NULL) // pDel is not the predecessor

1. parent = pDel

2. pDel = pDel->right

4. subroot->data = pDel->data5. if (parent = subroot)

1. parent->left = pDel->left

6. else

1. parent->right = pDel->left7. recycle pDel

End RemoveNode

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subroot

pDel

parent


(d)


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1. parent = subroot



1. parent = pDel




6. else


End RemoveNode

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subroot

pDel

parent


(d)


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1. parent = subroot



1. parent = pDel




6. else


End RemoveNode

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subroot

pDel

parent


(d)


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1. parent = subroot



1. parent = pDel




6. else


End RemoveNode

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subroot

pDel

parent


22 (d)


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1. parent = subroot



1. parent = pDel




6. else


End RemoveNode

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subroot

pDel

parent


22

22

(d)


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1. parent = subroot



1. parent = pDel




6. else


End RemoveNode

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parent

subroot

pDel

22

(d)

Performance of random BST

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The average number of nodes visited during a search of average BST

with n nodes approximately 2 ln2 = (2 ln 2) (lg n) 1.39 lg n

The average BST requires approximately 2 ln2 1.39 times as many

comparisons as a completely balanced tree.

Chapter+7+-+Tree+-+2009-+2009

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