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Unit-III
Tree
In linear data structure data is organized in sequential order and in non-linear data structure data is organized in random order. A tree is a very popular non-linear data structure used in a 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 recursively 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. Ex:
Terminology
In a tree data structure, we use the following terminology... Root In a tree data structure, the first node is called as Root Node. Every tree must have a root node. We can say that the root node is the origin of the tree data structure. In any tree, there must be only one root node. We never have multiple root nodes in a tree.
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. Parent In a tree data structure, the node which is a predecessor of any node is called as PARENT NODE. In simple words, the node which has a branch from it to any other node is called a parent node. Parent node can also be defined as " The node which has child / children ". 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.
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 ' 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). Height In a tree data structure, the total number of edges 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'.
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'. 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. Sub Tree In a tree data structure, each child from a node forms a subtree recursively. Every child node will form a subtree on its parent node.
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 2level^ number of nodes. For example at level 2 there must be 2^2 = 4 nodes and at level 3 there must be 2^3 = 8 nodes. - A binary tree in which every internal node has exactly two children and all leaf nodes are at same level is called Complete Binary Tree. Complete binary tree is also called as **Perfect Binary Tree
- Extended Binary Tree** A binary tree can be converted into Full Binary tree by adding dummy nodes to existing nodes wherever required.
- The full binary tree obtained by adding dummy nodes to a binary tree is called as Extended Binary Tree. In above figure, a normal binary tree is converted into full binary tree by adding dummy nodes (In pink colour).
Binary Tree Representations
A binary tree data structure is represented using two methods. Those methods are as follows...
- Array Representation
- Linked List Representation Consider the following binary tree...
1. Array Representation of Binary Tree In array representation of a binary tree, we use one-dimensional array (1-D Array) to represent a binary tree. Consider the above example of a binary tree and it is represented as follows... To represent a binary tree of depth 'n' using array representation, we need one dimensional array with a maximum size of 2n + 1. 2. Linked List Representation of Binary Tree We use a double linked list to represent a binary tree. In a double linked list, every node consists of three fields. First field for storing left child address, second for storing actual data and third for storing right child address. In this linked list representation, a node has the following structure... The above example of the binary tree represented using Linked list representation is shown as follows...
2. Pre - Order Traversal ( root - leftChild - rightChild ) In Pre-Order traversal, the root node is visited before the 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. In the above example of binary tree, first we visit root node 'A' then visit its left child 'B' which is a root for D and F. So we visit B's left child 'D' and again D is a root for I and J. So we visit D's left child 'I' which is the leftmost child. So next we go for visiting D's right child 'J'. With this we have completed root, left and right parts of node D and root, left parts of node B. Next visit B's right child 'F'. With this we have completed root and left parts of node A. So we go for A's right child 'C' which is a root node for G and H. After visiting C, we go for its left child 'G' which is a root for node K. So next we visit left of G, but it does not have left child so we go for G's right child 'K'. With this, we have completed node C's root and left parts. Next visit C's right child 'H' which is the rightmost child in the tree. So we stop the process. That means here we have visited in the order of A-B-D-I-J-F-C-G-K-H using Pre-Order Traversal. - Pre-Order Traversal for above example binary tree is A - B - D - I - J - F - C - G - K - H 3. 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. - Post-Order Traversal for above example binary tree is I - J - D - F - B - K - G - H - C – A
//Binary Tree Display using In-Order Traversals #include #include struct Node{ int data; struct Node left; struct Node right; }; struct Node root = NULL; int count = 0; struct Node insert(struct Node, int); void display(struct Node); void main(){ int choice, value; clrscr(); printf("\n----- Binary Tree -----\n"); while(1){ printf("\n***** MENU *****\n"); printf("1. Insert\n2. Display\n3. Exit"); printf("\nEnter your choice: "); scanf("%d",&choice); switch(choice){ case 1: printf("\nEnter the value to be insert: "); scanf("%d", &value); root = insert(root,value); break; case 2: display(root); break; case 3: exit(0); default: printf("\nPlease select correct operations!!!\n"); } } } struct Node* insert(struct Node *root,int value){ struct Node newNode; newNode = (struct Node)malloc(sizeof(struct Node)); newNode->data = value; if(root == NULL){ newNode->left = newNode->right = NULL; root = newNode; count++; } else{ if(count%2 != 0) root->left = insert(root->left,value); else root->right = insert(root->right,value); } return root; } // display is performed by using Inorder Traversal void display(struct Node *root) { if(root != NULL){ display(root->left); printf("%d\t",root->data); display(root->right); } }
- Every binary search tree is a binary tree but every binary tree need not to be binary search tree. Operations on a Binary Search Tree The following operations are performed on a binary search tree...
- Search
- Insertion
- Deletion Search Operation in BST In a binary search tree, the search operation is performed with O(log n) time complexity. The search operation is performed as follows... Step 1 - Read the search element from the user. Step 2 - Compare the search element with the value of root node in the tree. Step 3 - If both are matched, then display "Given node is found!!!" and terminate the function Step 4 - If both are not matched, then check whether search element is smaller or larger than that node value. Step 5 - If search element is smaller, then continue the search process in left subtree. Step 6- If search element is larger, then continue the search process in right subtree. Step 7 - Repeat the same until we find the exact element or until the search element is compared with the leaf node Step 8 - If we reach to the node having the value equal to the search value then display "Element is found" and terminate the function. Step 9 - If we reach to the leaf node and if it is also not matched with the search element, then display "Element is not found" and terminate the function. Insertion Operation in BST In a binary search tree, the insertion operation is performed with O(log n) time complexity. In binary search tree, new node is always inserted as a leaf node. The insertion operation is performed as follows... Step 1 - Create a newNode with given value and set its left and right to NULL. Step 2 - Check whether tree is Empty. Step 3 - If the tree is Empty , then set root to newNode. Step 4 - If the tree is Not Empty , then check whether the value of newNode is smaller or larger than the node (here it is root node). Step 5 - If newNode is smaller than or equal to the node then move to its left child. If newNode is larger than the node then move to its right child. Step 6- Repeat the above steps until we reach to the leaf node (i.e., reaches to NULL). Step 7 - After reaching the leaf node, insert the newNode as left child if the newNode is smaller or equal to that leaf node or else insert it as right child.
Deletion Operation in BST In a binary search tree, the deletion operation is performed with O(log n) time complexity. Deleting a node from Binary search tree includes following three cases... Case 1: Deleting a Leaf node (A node with no children) Case 2: Deleting a node with one child Case 3: Deleting a node with two children Case 1: Deleting a leaf node We use the following steps to delete a leaf node from BST... Step 1 - Find the node to be deleted using search operation Step 2 - Delete the node using free function (If it is a leaf) and terminate the function. Case 2: Deleting a node with one child We use the following steps to delete a node with one child from BST... Step 1 - Find the node to be deleted using search operation Step 2 - If it has only one child then create a link between its parent node and child node. Step 3 - Delete the node using free function and terminate the function. Case 3: Deleting a node with two children We use the following steps to delete a node with two children from BST... Step 1 - Find the node to be deleted using search operation Step 2 - If it has two children, then find the largest node in its left subtree (OR) the smallest node in its right subtree. Step 3 - Swap both deleting node and node which is found in the above step. Step 4 - Then check whether deleting node came to case 1 or case 2 or else goto step 2 Step 5 - If it comes to case 1 , then delete using case 1 logic. Step 6- If it comes to case 2 , then delete using case 2 logic. Step 7 - Repeat the same process until the node is deleted from the tree. Ex: Construct a Binary Search Tree by inserting the following sequence of numbers... 10,12,5,4,20,8,7,15 and 13 Above elements are inserted into a Binary Search Tree as follows...
//Binary Search Tree Implementation #include #include #include struct node { int data; struct node *left; struct node *right; }; void inorder(struct node *root) { if(root) { inorder(root->left); printf(" %d",root->data); inorder(root->right); } } int main() { int n , i; struct node *p , *q , *root; printf("Enter the number of nodes to be insert: "); scanf("%d",&n); printf("\nPlease enter the numbers to be insert: "); for(i=0;idata); p->left = NULL; p->right = NULL; if(i == 0) { root = p; // root always point to the root node } else { q = root; // q is used to traverse the tree while(1) { if(p->data > q->data) { if(q->right == NULL) { q->right = p; break; } else q = q->right; } else { if(q->left == NULL) { q->left = p; break; } else q = q->left; } } } } printf("\nBinary Search Tree nodes in Inorder Traversal: "); inorder(root); printf("\n"); return 0; }
AVL Tree
AVL tree is a height-balanced binary search tree. That means, an AVL tree is also a binary search tree but it is a balanced tree. A binary tree is said to be balanced if, the difference between the heights of left and right subtrees of every node in the tree is either -1, 0 or +1. In other words, a binary tree is said to be balanced if the height of left and right children of every node differ by either -1, 0 or +1. In an AVL tree, every node maintains an extra information known as balance factor. The AVL tree was introduced in the year 1962 by G.M. Adelson-Velsky and E.M. Landis. An AVL tree is defined as follows...
- An AVL tree is a balanced binary search tree. In an AVL tree, balance factor of every node is either -1, 0 or +1. Balance factor of a node is the difference between the heights of the left and right subtrees of that node. The balance factor of a node is calculated either height of left subtree - height of right subtree (OR) height of right subtree - height of left subtree. In the following explanation, we calculate as follows... Balance factor = heightOfLeftSubtree – heightOfRightSubtree Ex: The above tree is a binary search tree and every node is satisfying balance factor condition. So this tree is said to be an AVL tree.
- Every AVL Tree is a binary search tree but every Binary Search Tree need not be AVL tree. AVL Tree Rotations In AVL tree, after performing operations like insertion and deletion we need to check the balance factor of every node in the tree. If every node satisfies the balance factor condition then we conclude the operation otherwise we must make it balanced. Whenever the tree becomes imbalanced due to any operation we use rotation operations to make the tree balanced. Rotation operations are used to make the tree balanced.
- Rotation is the process of moving nodes either to left or to right to make the tree balanced. There are four rotations and they are classified into two types.
Operations on an AVL Tree The following operations are performed on AVL tree...
- Search
- Insertion
- Deletion Search Operation in AVL Tree In an AVL tree, the search operation is performed with O(log n) time complexity. The search operation in the AVL tree is similar to the search operation in a Binary search tree. We use the following steps to search an element in AVL tree... Step 1 - Read the search element from the user. Step 2 - Compare the search element with the value of root node in the tree. Step 3 - If both are matched, then display "Given node is found!!!" and terminate the function Step 4 - If both are not matched, then check whether search element is smaller or larger than that node value. Step 5 - If search element is smaller, then continue the search process in left subtree. Step 6 - If search element is larger, then continue the search process in right subtree. Step 7 - Repeat the same until we find the exact element or until the search element is compared with the leaf node. Step 8 - If we reach to the node having the value equal to the search value, then display "Element is found" and terminate the function. Step 9 - If we reach to the leaf node and if it is also not matched with the search element, then display "Element is not found" and terminate the function. Insertion Operation in AVL Tree In an AVL tree, the insertion operation is performed with O(log n) time complexity. In AVL Tree, a new node is always inserted as a leaf node. The insertion operation is performed as follows... Step 1 - Insert the new element into the tree using Binary Search Tree insertion logic. Step 2 - After insertion, check the Balance Factor of every node. Step 3 - If the Balance Factor of every node is 0 or 1 or -1 then go for next operation. Step 4 - If the Balance Factor of any node is other than 0 or 1 or -1 then that tree is said to be imbalanced. In this case, perform suitable Rotation to make it balanced and go for next operation. Ex: Construct an AVL Tree by inserting numbers from 1 to 8.
