Tree Concept in Data Structure

Name : Uday Shah - HOD (IT)
Contact No : 7600044051
E-Mail : rupareleducation@gmail.com
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Tree Terminology

In linear data structure, data is organized in sequential order and in non-linear data structure, data is organized in random order. Tree is a very popular data structure used in wide range of applications. A tree data structure can be defined as follows...

Tree is a non-linear data structure which organizes data in hierarchical structure and this is a recursive definition.

A tree data structure can also be defined as follows...

Tree data structure is a collection of data (Node) which is organized in hierarchical structure and this is a recursive definition

In tree data structure, every individual element is called as Node. Node in a tree data structure, stores the actual data of that particular element and link to next element in hierarchical structure.

In a tree data structure, if we have N number of nodes then we can have a maximum of N-1 number of links.

Example

Terminology

In a tree data structure, we use the following terminology...

1. Root

In a tree data structure, the first node is called as Root Node. Every tree must have root node. We can say that root node is the origin of tree data structure. In any tree, there must be only one root node. We never have multiple root nodes in a tree.

2. Edge

In a tree data structure, the connecting link between any two nodes is called as EDGE. In a tree with 'N' number of nodes there will be a maximum of 'N-1' number of edges.

3. Parent

In a tree data structure, the node which is predecessor of any node is called as PARENT NODE. In simple words, the node which has branch from it to any other node is called as parent node. Parent node can also be defined as "The node which has child / children".

4. Child

In a tree data structure, the node which is descendant of any node is called as CHILD Node. In simple words, the node which has a link from its parent node is called as child node. In a tree, any parent node can have any number of child nodes. In a tree, all the nodes except root are child nodes.

5. Siblings

In a tree data structure, nodes which belong to same Parent are called as SIBLINGS. In simple words, the nodes with same parent are called as Sibling nodes.

6. Leaf

In a tree data structure, the node which does not have a child is called as LEAF Node. In simple words, a leaf is a node with no child. 

In a tree data structure, the leaf nodes are also called as External Nodes. External node is also a node with no child. In a tree, leaf node is also called as 'Terminal' node.

7. Internal Nodes

In a tree data structure, the node which has atleast one child is called as INTERNAL Node. In simple words, an internal node is a node with atleast one child. 

In a tree data structure, nodes other than leaf nodes are called as Internal Nodes. The root node is also said to be Internal Node if the tree has more than one node. Internal nodes are also called as 'Non-Terminal' nodes.

8. Degree

In a tree data structure, the total number of children of a node is called as DEGREE of that Node. In simple words, the Degree of a node is total number of children it has. The highest degree of a node among all the nodes in a tree is called as 'Degree of Tree'

9. Level

In a tree data structure, the root node is said to be at Level 0 and the children of root node are at Level 1 and the children of the nodes which are at Level 1 will be at Level 2 and so on... In simple words, in a tree each step from top to bottom is called as a Level and the Level count starts with '0' and incremented by one at each level (Step).

10. Height

In a tree data structure, the total number of egdes from leaf node to a particular node in the longest path is called as HEIGHT of that Node. In a tree, height of the root node is said to be height of the tree. In a tree, height of all leaf nodes is '0'.

11. Depth

In a tree data structure, the total number of egdes from root node to a particular node is called as DEPTH of that Node. In a tree, the total number of edges from root node to a leaf node in the longest path is said to be Depth of the tree. In simple words, the highest depth of any leaf node in a tree is said to be depth of that tree. In a tree, depth of the root node is '0'.

12. Path

In a tree data structure, the sequence of Nodes and Edges from one node to another node is called as PATH between that two Nodes. Length of a Path is total number of nodes in that path. In below example the path A - B - E - J has length 4.

Binary Tree

In a normal tree, every node can have any number of children. Binary tree is a special type of tree data structure in which every node can have a maximum of 2 children. One is known as left child and the other is known as right child.

A tree in which every node can have a maximum of two children is called as Binary Tree.

In a binary tree, every node can have either 0 children or 1 child or 2 children but not more than 2 children.

Example

There are different types of binary trees and they are...

1. Strictly Binary Tree

In a binary tree, every node can have a maximum of two children. But in strictly binary tree, every node should have exactly two children or none. That means every internal node must have exactly two children. A strictly Binary Tree can be defined as follows...

A binary tree in which every node has either two or zero number of children is called Strictly Binary Tree

Strictly binary tree is also called as Full Binary Tree or Proper Binary Tree or 2-Tree

Strictly binary tree data structure is used to represent mathematical expressions.

2. Complete Binary Tree

In a binary tree, every node can have a maximum of two children. But in strictly binary tree, every node should have exactly two children or none and in complete binary tree all the nodes must have exactly two children and at every level of complete binary tree there must be 2^level number of nodes. For example at level 2 there must be 2² = 4 nodes and at level 3 there must be 2³ = 8 nodes.

 Binary Tree Traversals

When we wanted to display a binary tree, we need to follow some order in which all the nodes of that binary tree must be displayed. In any binary tree displaying order of nodes depends on the traversal method.

Displaying (or) visiting order of nodes in a binary tree is called as Binary Tree Traversal.

There are three types of binary tree traversals.

1.   In - Order Traversal

2.   Pre - Order Traversal

3.   Post - Order Traversal

Consider the following binary tree...

1. In - Order Traversal ( leftChild - root - rightChild )

In In-Order traversal, the root node is visited between left child and right child. In this traversal, the left child node is visited first, then the root node is visited and later we go for visiting right child node. This in-order traversal is applicable for every root node of all subtrees in the tree. This is performed recursively for all nodes in the tree.

That means here we have visited in the order of I - D - J - B - F - A - G - K - C - H using In-Order Traversal.

2. Pre - Order Traversal ( root - leftChild - rightChild )

In Pre-Order traversal, the root node is visited before left child and right child nodes. In this traversal, the root node is visited first, then its left child and later its right child. This pre-order traversal is applicable for every root node of all subtrees in the tree. 

That means here we have visited in the order of A-B-D-I-J-F-C-G-K-H using Pre-Order Traversal.

2. Post - Order Traversal ( leftChild - rightChild - root )

In Post-Order traversal, the root node is visited after left child and right child. In this traversal, left child node is visited first, then its right child and then its root node. This is recursively performed until the right most node is visited.

Here we have visited in the order of I - J - D - F - B - K - G - H - C - A using Post-Order Traversal.

Double Linked List Data Structure for BCA

Name : Uday Shah HOD- IT
Contact No : 7600044051
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Double Linked List

What is Double Linked List?

In a single linked list, every node has link to its next node in the sequence. So, we can traverse from one node to other node only in one direction and we can not traverse back. We can solve this kind of problem by using double linked list. Double linked list can be defined as follows...

Double linked list is a sequence of elements in which every element has links to its previous element and next element in the sequence.

In double linked list, every node has link to its previous node and next node. So, we can traverse forward by using next field and can traverse backward by using previous field. Every node in a double linked list contains three fields and they are shown in the following figure...

Here, 'link1' field is used to store the address of the previous node in the sequence, 'link2' field is used to store the address of the next node in the sequence and 'data' field is used to store the actual value of that node.

Example

☀ In double linked list, the first node must be always pointed by head.
☀ Always the previous field of the first node must be NULL.
☀ Always the next field of the last node must be NULL.

Operations

In a double linked list, we perform the following operations...

1.    Insertion

2.    Deletion

3.    Display

Insertion

In a double linked list, the insertion operation can be performed in three ways as follows...

1.    Inserting At Beginning of the list

2.    Inserting At End of the list

3.    Inserting At Specific location in the list

Inserting At Beginning of the list

We can use the following steps to insert a new node at beginning of the double linked list...

• Step 1: Create a newNode with given value and newNode → previous as NULL.
• Step 2: Check whether list is Empty (head == NULL)
• Step 3: If it is Empty then, assign NULL to newNode → next and newNode to head.
• Step 4: If it is not Empty then, assign head to newNode → next and newNode to head.

Inserting At End of the list

We can use the following steps to insert a new node at end of the double linked list...

• Step 1: Create a newNode with given value and newNode → next as NULL.
• Step 2: Check whether list is Empty (head == NULL)
• Step 3: If it is Empty, then assign NULL to newNode → previous and newNode to head.
• Step 4: If it is not Empty, then, define a node pointer temp and initialize with head.
• Step 5: Keep moving the temp to its next node until it reaches to the last node in the list (until temp → next is equal to NULL).
• Step 6: Assign newNode to temp → next and temp to newNode → previous.

Inserting At Specific location in the list (After a Node)

We can use the following steps to insert a new node after a node in the double linked list...

• Step 1: Create a newNode with given value.
• Step 2: Check whether list is Empty (head == NULL)
• Step 3: If it is Empty then, assign NULL to newNode → previous & newNode → next and newNode to head.
• Step 4: If it is not Empty then, define two node pointers temp1 & temp2 and initialize temp1 with head.
• Step 5: Keep moving the temp1 to its next node until it reaches to the node after which we want to insert the newNode (until temp1 → data is equal to location, here location is the node value after which we want to insert the newNode).
• Step 6: Every time check whether temp1 is reached to the last node. If it is reached to the last node then display 'Given node is not found in the list!!! Insertion not possible!!!' and terminate the function. Otherwise move the temp1 to next node.
• Step 7: Assign temp1 → next to temp2, newNode to temp1 → next, temp1 to newNode → previous, temp2 to newNode → next and newNode to temp2 → previous.

Deletion

In a double linked list, the deletion operation can be performed in three ways as follows...

1.    Deleting from Beginning of the list

2.    Deleting from End of the list

3.    Deleting a Specific Node

Deleting from Beginning of the list

We can use the following steps to delete a node from beginning of the double linked list...

• Step 1: Check whether list is Empty (head == NULL)
• Step 2: If it is Empty then, display 'List is Empty!!! Deletion is not possible' and terminate the function.
• Step 3: If it is not Empty then, define a Node pointer 'temp' and initialize with head.
• Step 4: Check whether list is having only one node (temp → previous is equal to temp → next)
• Step 5: If it is TRUE, then set head to NULL and delete temp (Setting Empty list conditions)
• Step 6: If it is FALSE, then assign temp → next to head, NULL to head → previous and delete temp.

Deleting from End of the list

We can use the following steps to delete a node from end of the double linked list...

• Step 1: Check whether list is Empty (head == NULL)
• Step 2: If it is Empty, then display 'List is Empty!!! Deletion is not possible' and terminate the function.
• Step 3: If it is not Empty then, define a Node pointer 'temp' and initialize with head.
• Step 4: Check whether list has only one Node (temp → previous and temp → next both are NULL)
• Step 5: If it is TRUE, then assign NULL to head and delete temp. And terminate from the function. (Setting Empty list condition)
• Step 6: If it is FALSE, then keep moving temp until it reaches to the last node in the list. (until temp → next is equal to NULL)
• Step 7: Assign NULL to temp → previous → next and delete temp.

Deleting a Specific Node from the list

We can use the following steps to delete a specific node from the double linked list...

• Step 1: Check whether list is Empty (head == NULL)
• Step 2: If it is Empty then, display 'List is Empty!!! Deletion is not possible' and terminate the function.
• Step 3: If it is not Empty, then define a Node pointer 'temp' and initialize with head.
• Step 4: Keep moving the temp until it reaches to the exact node to be deleted or to the last node.
• Step 5: If it is reached to the last node, then display 'Given node not found in the list! Deletion not possible!!!' and terminate the fuction.
• Step 6: If it is reached to the exact node which we want to delete, then check whether list is having only one node or not
• Step 7: If list has only one node and that is the node which is to be deleted then set head to NULL and delete temp (free(temp)).
• Step 8: If list contains multiple nodes, then check whether temp is the first node in the list (temp == head).
• Step 9: If temp is the first node, then move the head to the next node (head = head → next), set head of previous to NULL (head → previous = NULL) and delete temp.
• Step 10: If temp is not the first node, then check whether it is the last node in the list (temp → next == NULL).
• Step 11: If temp is the last node then set temp of previous of next to NULL (temp → previous → next = NULL) and delete temp (free(temp)).
• Step 12: If temp is not the first node and not the last node, then set temp of previous of next to temp of next (temp → previous → next = temp → next), temp of next of previous to temp of previous (temp → next → previous = temp → previous) and delete temp (free(temp)).

Displaying a Double Linked List

We can use the following steps to display the elements of a double linked list...

• Step 1: Check whether list is Empty (head == NULL)
• Step 2: If it is Empty, then display 'List is Empty!!!' and terminate the function.
• Step 3: If it is not Empty, then define a Node pointer 'temp' and initialize with head.
• Step 4: Display 'NULL <--- '.
• Step 5: Keep displaying temp → data with an arrow (<===>) until temp reaches to the last node
• Step 6: Finally, display temp → data with arrow pointing to NULL (temp → data ---> NULL).