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TREES

General trees

Binary trees

Binary search trees AVL trees

Balanced and Threaded trees.

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General Trees.Trees can be defined in two ways :

Recursive

Non- recursive A

B C D E

F G H I

K

J

One natural way to define a tree is

recursively

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General trees-Definition A tree is a collection of nodes. The

collection can be empty; otherwise atree consists of a distinguish node r,called root, and zero or more non-

empty (sub)trees T1,T2,T3,.TK. .

A

B C D E

F G H I

K

J

Each of whose roots areconnected by a directed edge

from r.

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General trees-Definition

A tree consists of set of nodes andset of edges that connected pair ofnodes.

A

B C D E

F G HI

K

J

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Eg. A table of contents and

its tree representationBookC1

S1.1

S1.2C2

S2.1

S2.1.1

S2.1.2

S2.2

S2.3

C3

Book

C1 C2 C3

S2.1S1.2S1.1 S2.2 S2.3

S2.1.2S2.1.1

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Degree

The number of sub tree

of a node is called its

degree.

Eg. Degree of book

3, C12,C30

Book

C1 C2 C3

S2.1S1.2 S2.2 S2.3

S2.1.2S2.1.1

S1.1

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Terminal Nodes and Non

Terminal nodesNodes that have degree 0 is calledLeaf or Terminal node.Other nodescalled non-terminal nodes. Eg.Leaf

nodes :C3,S1.1,S1.2 etc.

Book

C1 C2 C3

S2.1S1.2 S2.2S2.3

S2.1.2S2.1.1

S1.1

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Parent, Children & SiblingsBook is said to be the father (parent)of C1,C2,C3 and C1,C2,C3 are said

to be sons (children ) of book.Children of the same parent are said

to be siblings.Eg.C1,C2,C3 are siblings (Brothers)

Book

C1 C2 C3

S2.1S1.2 S2.2 S2.3

S2.1.2S2.1.1

S1.1

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Length

Book

C1 C2 C3

S2.1S1.2 S2.2 S2.3

S2.1.2S2.1.1

The length of a path is oneless than the number of nodesin the path.(Eg path from book

to s1.1=3-1=2)

S1.1

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Ancestor & Descendent

Book

C1 C2 C3

S2.1S1.2 S2.2 S2.3

S2.1.2S2.1.1

If there is a path from node a to node b, then a is an ancestor of b and b is

descendent of a.In above example, the ancestor of S2.1

are itself,C2 and book, while itdescendent are itself, S2.1.1 and S2.1.2.

S1.1

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Height & DepthThe height of a node in a tree is the length of

a longest path from node to leaf.[ In above

example node C1 has height 1, node C2 hasheight 2.etc.

Depth : The depth of a tree is the maximumlevel of any leaf in the tree.[ In above example

depth=3]

Book

C1 C2 C3

S2.1S1.2 S2.2 S2.3

S2.1.2S2.1.1

S1.1

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Tree -

Implementation : Keep the children of each node in a

linked list of tree nodes. Thus eachnode keeps two references : one toits leftmost child and other one for itsright sibling.

Left DataRight

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Left child -Right sibling

representation of a treeA

B

C

D E

GF

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A

B C D

F G

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Data structure

definition

Class Treenode

{

Object element;

Treenode leftchild;

Treenode rightsibling;

}

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An application :File system

There are many applications for

trees. A popular one is thedirectory structure in manycommon operating systems,including VAX/VMX,Unix and DOS.

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Binary trees

A binary tree is a tree in which nonodes can have more than twochildren.

The recursive definition is that abinary tree is either empty orconsists of a root, a left tree, and aright tree.

The left and right trees maythemselves be empty; thus a nodewith one child could have a left orright child. We use the recursive

definition several times in the

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One use of the binary tree is inthe expression tree, which iscentral data structure incompiler design.

- d

b c

+

a *

(a+((b-c)*d))

The leaves of an expression tree are

operands, such as constant, variable

names. The other nodes contain

operators.

Eg :

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Tree traversal-iterateclasses.

The main tree traversal

techniques are:Pre-order traversal

In-order traversalPost-order traversal

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Pre-order traversalTo traverse a non-emptybinary tree in pre-order (alsoknown as depth first order),we perform the followingoperations.Visit the root ( or print the root)Traverse the left in pre-order(Recursive)Traverse the right tree in pre-order

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Pre-order traversal

Pre-order list 1,2,3,5,8,9,6,10,4,7

1

2 43

5 6

8 9 10

7

Visit the root ( or print the root)Traverse the left in pre-order (Recursive)Traverse the right tree in pre-order

(Recursive)

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In-order traversal

1

243

5 6 7

8 9 10

Pre-order list 2,1,8,5,9,3,10,6,7,4

Traverse the left-subtree in in-order

Visit the rootTraverse the right-subtree in in-

order.

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post-order

traversal1

2 43

5 6 7

89 10

Pre-order list 2,8,9,5,10,6,3,7,4,1

Traverse the left sub-tree in post-orderTraverse the right sub-tree in post-order

Visit the root

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Properties of binary

trees.If a binary tree contains m nodes atlevel L, then it contains at most 2mnodes at level L+1.

A binary tree can contain at most 2L

nodes at L

At level 0 B-tree can contain at most 1=20 nodes

At level 1 B-tree can contain at most 2=21 nodes

At level 2 B-tree can contain at most 4=

22

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Complete B-tree

A complete B-tree of depth d is theB-tree that contains exactly 2Lnodes at each level between 0 and

d ( or 2d

nodes at d)

Complete B-treeNot a Complete B-tree

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The total number of nodes (Tn)

in a complete binary tree ofdepth d is 2d+1-1

Tn=20+21+22+2d..(1)

2Tn=21

+22

+ 2d+1

.(2) (2)-(1)

Tn=2d+1-1

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Threaded Binary TreesGiven a binary tree with n nodes, the total number of links in the tree is 2n.

Each node (except the root) has exactly one incoming arc only n - 1 links point to nodes remaining n + 1 links are null.

One can use these null links to simplify some traversal processes.

A threaded binary search tree is a BST with unused links employed to

point to other tree nodes. Traversals (as well as other operations, such as backtracking)

made more efficient.

A BST can be threaded with respect to inorder, preorder or postorder

successors.

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Threaded Binary TreesGiven the following BST, thread it to facilitate inorder traversal:

The first node visited in an inorder traversal is the leftmost leaf, node A.

Since A has no right child, the next node visited in this traversal is its

parent, node B.

Use the right pointer of node A as a thread to parent B to make backtracking

easy.

H

E

B

K

F J L

A D M

C

IG

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Threaded Binary Trees

The thread from A to B is shown as the arrow in above diagram.

H

E

B

K

F J L

A D M

C

IG

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Threaded Binary TreesThe next node visited is C, and since its right pointer is null, it also can be

used as a thread to its parent D:

H

E

B

K

F J L

A D M

C

IG

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Threaded Binary TreesNode D has a null right pointer which can be replaced with a pointer to Ds

inorder successor, node E:

H

E

B

K

F J L

A D M

C

IG

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Threaded Binary TreesThe next node visited with a null right pointer is G. Replace the null pointer

with the address of Gs inorder successor: H

H

E

B

K

F J L

A D M

C

IG

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Threaded Binary TreesFinally, we replace:

first, the null right pointer of I with a pointer to its parentand then, likewise, the null right pointer of J with a pointer to its parent

H

E

B

K

F J L

A D M

C

IG

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Threaded Tree Example

Amir Kamil 8/8/02 34

8

753

11

13

1

6

9

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Threaded Tree Traversal

We start at the leftmost node in thetree, print it, and follow its rightthread

If we follow a thread to the right, weoutput the node and continue to itsright

If we follow a link to the right, we goto the leftmost node, print it, andcontinue


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8

75

3

11

13

1

6

9Start at leftmost node, print it

Output1

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8

75

3

11

13

1

6

9Follow thread to right, print node

Output1

3

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8

75

3

11

13

1

6

9Follow link to right, go to

leftmost node and print

Output1

3

5

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8

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13

1

6


Output1

3

5

6

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8

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1

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Output1

3

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6

7

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8

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1

6


Output1

3

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7

8

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8

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1

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Output1

3

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8

9

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Output1

3

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7

8

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11

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Output1

3

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6

7

8

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13

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Threaded Tree Modification

Were still wasting pointers, sincehalf of our leafs pointers are still null

We can add threads to the previousnode in an inorder traversal as well,which we can use to traverse thetree backwards or even to do

postorder traversals


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Threaded Tree Modification

8

75

3

11

13

1

6

9

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