unit 3&5 trees

Session: Jan – June 2013

EEC 012 - DATA STRUCTURES – TREES 1Prof. (Dr) N Guruprasad

UNIT 3 & 5 – Trees & Search UNIT 3 & 5 – Trees & Search TreesTrees



Agenda

1) Introduction & Basic Terminologies2) Complete & Extended Binary tree3) Tree Traversal Algorithms4) BST and primitive operations,5) Threaded Binary tree and traversal6) Huffman Algorithm7) Heap sort8) AVL tree9) Introduction to m-way Search Trees, B Trees & B+ trees10) Storage Management: Garbage Collection



Introduction & definition

• One of the most important data structures in CS.• A tree consists of finite set of elements, called nodes,

and a finite set of directed lines called branches, that connect the nodes.

• Non linear data structure capable of expressing more complex relationship than that of physical adjacency.



Applications

1) Manipulate hierarchical data.2) Make information easy to search (traversal).3) Manipulate sorted lists of data.4) As a workflow for composing digital images for visual effects.5) Router algorithms.



Basic Tree Terminologies• Degree The number of branches associated with a node

is the degree of the node.• Edge • Path • Branch • Terminal node • Level number • Height / depth



Basic Tree Terminologies• Internal node node that is not a root or a leaf is known

as an internal node.• When the branch is directed toward the node, it is indegree

branch.• When the branch is directed away from the node, it is an

outdegree branch.• The sum of the indegree and outdegree branches is the

degree of the node.• If the tree is not empty, the first node is called the root.



Binary Trees

• A binary tree is a tree in which no node can have more than two subtrees; the maximum outdegree for a node is two.

• In other words, a node can have zero, one, or two subtrees.• These subtrees are designated as the left subtree and the

right subtree.



A null tree is a tree with no nodes



Some Properties of Binary Trees

• The height of binary trees can be mathematically predicted

• Given that we need to store N nodes in a binary tree, the maximum height is

maxH NA tree with a maximum height is rare. It occurs when all of the nodes in the entire tree have only one successor.




• The minimum height of a binary tree is determined as follows:

min 2log 1H N

For instance, if there are three nodes to be stored in the binary tree (N=3) then Hmin=2.




• Given a height of the binary tree, H, the minimum number of nodes in the tree is given as follows:

minN H




• The formula for the maximum number of nodes is derived from the fact that each node can have only two descendents. Given a height of the binary tree, H, the maximum number of nodes in the tree is given as follows:

max 2 1HN




• The children of any node in a tree can be accessed by following only one branch path, the one that leads to the desired node.

• The nodes at level 1, which are children of the root, can be accessed by following only one branch; the nodes of level 2 of a tree can be accessed by following only two branches from the root, etc.

• The balance factor of a binary tree is the difference in height between its left and right subtrees:

L RB H H



B=0 B=0 B=1 B=-1

B=0 B=1

B=-2 B=2

Balance of the tree



Complete and nearly complete binary trees

• A complete tree has the maximum number of entries for its height. The maximum number is reached when the last level is full.

• A tree is considered nearly complete if it has the minimum height for its nodes and all nodes in the last level are found on the left



Representation of a Binary Tree

• There are two ways of representing tree in memory :1) Linked representation2) Sequential representation



Linked Representation :

• We use three arrays :1) INFO[K] contains data at node N2) LEFT[K] contains location of left child of N3) RIGHT [K] contains location of right child of N

ROOT will contain location of root R of T.



• Consider the binary tree and represent it in linked representation format.



Sequential Representation :

• Uses single linear array called TREE :1) ROOT node R is stored in TREE[1]2) LEFT node is placed at TREE[2*K] where K position of its parent node3) RIGHT node is placed at TREE[2*K+1] where K position of its parent node



• Consider the binary tree and represent it in sequential representation format.



Primitive operations

• Traversing• Insertion• Deletion



Binary Tree Traversal

• A binary tree traversal requires that each node of the tree be processed once and only once in a predetermined sequence.

• Three types :• Pre order traversal algorithm• In order traversal algorithm• Post order traversal algorithm



Preorder Traversal

Step 1 : Process root TStep 2 : Traverse left subtree again in PREORDERStep 3 : Traverse right subtree again in PREORDER

Also called NLR ( Node, Left, Right ) Traversal

Ex : A B D E F C G H J L K



Function representation :pre_trav(struct btree *tree) {

if(root==NULL)printf(“\n Tree is empty “);

getch();}

if(tree!=NULL) {printf(“\n %d “,treeinfo);pre_trav(treeleft);pre_trav(treeright);

}else return;



Inorder Traversal

Step 1 : Traverse left subtree in INORDERStep 2 : Process root TStep 3 : Traverse right subtree again in INORDER

Also called LNR (Left, Node, Right ) Traversal

Ex : D B F E A G C L J H K



Function representation :in_trav(struct btree *tree) {


getch();}

if(tree!=NULL) {in_trav(treeleft);printf(“\n %d “,treeinfo);in_trav(treeright);

}else return;



Postorder Traversal

Step 1 : Traverse left subtree in POSTORDERStep 2 : Traverse right subtree again in POSTORDERStep 3 : Process root T

Also called LRN (Left, Right, Node ) Traversal

Ex : D F E B G L J K H C A



Function representation :post_trav(struct btree *tree) {


getch();}

if(tree!=NULL) {post_trav(treeleft);post_trav(treeright);printf(“\n %d “,treeinfo);

}else return;



Binary Search TreesThis data structure enables us to search an element. Suppose T is a binary tree ,then T is called BINARY SEARCH TREE if each node N of T has the following properties :-

the value at any node N is greater than every value in left subtree and less than every value in right subtree.

Also called BINARY SORTED TREE



Example : ( Fig BST – 1 )

38

14 56

8 23 45 82

18 70



Primitive operations on BST

• Traversing• Insertion• Deletion



w.r.t fig BST-1

Pre order traversal :38 14 8 23 18 56 45 82 70

In order traversal :8 14 18 23 38 45 56 70 82

Post order traversal :8 18 23 14 45 70 82 56 38



Searching & Inserting in a BST

Suppose an item of information is given, the following algorithm finds the location of ITEM in binary search tree T or inserts ITEM as a new node in its appropriate place.

a) Compare ITEM with root node N

1) If ITEM<N, proceed to left subtree. 2) If ITEM>N, proceed to right subtree.



Searching & Inserting in a BST

b) Repeat step (a) until one of the following occurs, 1) We meet a node N such that ITEM=N. In this case,

search is successful and no insertion takes place.

2) We meet an empty subtree which indicates search is unsuccessful and ITEM is inserted in place of an empty subtree.



Example :

Consider fig BST – 1

Suppose ITEM = 20 is to be inserted.

Answer : We insert as right successor of 18



Deleting in a BST

There are three different types of deletion in a BST.1) Nodes with no successor (leaf node )2) Nodes with one successor 3) Nodes with two successors

Let us consider the tree T ( BST-2) and its linked representation



Example : ( Fig BST – 2 ) 60

2575

15 5066

33

44



INFO RIGHT LEFT

1 33 0 9

Root 2 25 8 10

3 3 60 2 7

4 66 0 0

5 6

Avail 6 0

5 7 75 4 0

8 15 0 0

9 44 0 0

10 50 1 0



1) Nodes with no successor (leaf node )

Ex : 15, 44 & 66Delete : 44Draw the tree and linked representation.



60

2575

15 5066

33



INFO RIGHT LEFT

1 33 0 9

Root 2 25 8 10

3 3 60 2 7

4 66 0 0

5 6

Avail 6 0

9 7 75 4 0

8 15 0 0

9 5

10 50 1 0



2) Nodes with one successor

Ex : 50, 33 & 75Delete : 75Draw the tree and linked representation.



60

2566

15 50

33

44



INFO RIGHT LEFT

1 33 0 9

Root 2 25 8 10

3 3 60 2 4

4 66 0 0

5 6

Avail 6 0

7 7 5 0

8 15 0 0

9 44 0 0

10 50 1 0



3) Nodes with two successor

Ex : 25 & 60Delete : 25Draw the tree and linked representation.



60

3375

15 5066

44



INFO RIGHT LEFT

1 33 8 10

Root 2 5

3 3 60 1 7

4 66 0 0

5 6

Avail 6 0

2 7 75 4 0

8 15 0 0

9 44 0 0

10 50 9 0



Problem Suppose the following eight numbers are inserted in order into an empty BST T.

50 33 44 22 77 35 60

Draw the tree T



Problem Suppose the following list of letters is inserted in order into an empty BST T.

J R D G T E M H P A F Q

Draw the tree T and also find INORDER traversal.

unit 3&5 trees

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