8. Write an example for expression tree. [A/M 10] (A+B)*((C-D)/(E^F))

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1 DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING EC6301 OBJECT ORIENTED PROGRAMMING AND DATA STRUCTURES UNIT IV NONLINEAR DATA STRUCTURES Part A 1. Define Tree [N/D 08] Tree is a non-liner data structure, which is used to store data item in a shorted sequence. It represents any hierarchical relationship between any data item. It is a collection of nodes, which has a distinguished node called root node and zero or more non-empty sub trees T1, T2.Tk, each of which are connected by a directed edge from the root. 2. Define Full Binary Tree [A/M 07] A full binary tree (sometimes 2-tree or strictly binary tree) is a tree in which every node other than the leaves has two children. A full tree is sometimes ambiguously defined as a perfect tree. Physicists define a binary tree to mean a full binary tree. A binary tree T is full if each node is either a leaf or possesses exactly two child nodes. 3. Define Complete Binary Tree [A/M 07] A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible. A tree is called an almost complete binary tree or nearly complete binary tree if the exception holds, i.e. the last level is not completely filled. This type of tree is used as a specialized data structure called a heap. A binary tree T with n levels is complete if all levels except possibly the last are completely full and the last level has all its nodes to the left side. 4. State the properties of a binary tree. [N/D 09] The properties of binary tree the properties of binary tree are: 1) A binary tree of n elements has n-1 edges 2) A binary tree of height h has at least h and at most 2h - 1 elements 3) The height of a binary tree with n elements is at most n and at least log2 (n+1) 5. Define Binary Tree [N/D 09] Binary tree is a finite set of elements that is either empty or is partitioned into three disjoint subsets. The first subset contains a single element called the root of the tree. The other two subsets are themselves binary trees called the left and right sub trees.

2 6. What is a threaded binary tree? [N/D 10] A Threaded Binary Tree is a binary tree in which every node that does not have a right child has a THREAD (in actual sense, a link) to its INORDER successor. By doing this threading we avoid the recursive method of traversing a Tree, which makes use of stacks and consumes a lot of memory and time. 7. Write an algorithm to declare node of a tree structure. [A/M 10] Algorithm to declare node of a tree structure: typedefstruct node int data; struct node *left; struct node *right; }bin; 8. Write an example for expression tree. [A/M 10] (A+B)*((C-D)/(E^F)) 9. List out the advantages of a threaded binary tree. [N/D 09] The advantages of a threaded binary tree are: 1) By doing threading we avoid the recursive method of traversing a Tree, which makes use of stack and consumes a lot of memory and time. 2) The node can keep record of its root. 10. Define Expression Tree [N/D 09] Expression tree is also a binary tree in which leaf nodes are terminal nodes or operands and non-terminal intermediate nodes are operators used for traversal. Example:

3 11. Differentiate Graph from Tree. [N/D 11] 1) A tree is a specialized case of a graph. A tree is a connected graph with no circuits and no self-loops. 2) A graph G consists of a nonempty set V which is a set of nodes of the graph, a set E which is the set of edges of the graph, and a mapping from the set for edge E to a set of pairs of elements of V. 12. Define Non-linear Data structure Non Linear data structures are data structures that have a linear relationship between its adjacent elements but it has the hierarchical relationship. Example: Graph, Tree. 13. List out the applications of tree data structure. The applications of tree data structure are: 1) On a computer - The file scheme uses tree architecture; a list of options system also has the same idea of selecting radical strata of options. 2) Indexes in a book have a narrow tree structure - letter/subjects starting with that letter. Contents probably have a superior layout - the different levels of chapters. 3) In computer science, a tree is a commonly used data formation that feigns a hierarchical tree structure with a set of linked nodes. 14. Define Leaf Nodes with no children are known as leaves. A leaf will always have degree zero and is also called terminal node. 15. What is node level? The level of a node is defined recursively by assuming the level of the root to be one and if a node is at level l, then its children at level l Define Degree of a Node It is the number of sub trees of a node in a given tree. 17. Define Siblings Nodes with the same parent are called siblings.

4 18. Define Height The height of a node n is the length of the longest path from root to a leaf. Thus all leaves have height zero. The height of a tree is equal to a height of a root. 19. What is meant by path length in a tree? The path length in tree is a number of edges between nodes from the root to the node. The depth of a tree is the length of the longest path from the root to leaf node. 20. Define Forest A forest is a disjoint union of trees. A forest is a collection on N (N>0) disjoint tree or group of trees. If the root is removed from the tree, that tree becomes a forest. 21. What is an ordered tree? An ordered tree is an oriented tree in which the children of a node are somehow "ordered." If T 1 and T 2 are ordered trees then T 1 T 2 else T 1 = T What are the applications of binary tree? The applications of binary tree are: 1) Used in almost every high-bandwidth router for storing router-tables. 2) Used in Huffman code construction. 3) Used in parse tree construction. 23. What is meant by traversing? Traversing a tree means processing it in such a way, that each node is visited only once. Traversal is an operation which can be performed on a binary tree is visiting all the nodes exactly once. 24. What are the different types of traversals? The different types of traversing are: 1) Pre-order traversal-yields prefix form of expression 2) In-order traversal-yields infix form of expression 3) Post-order traversal-yields postfix form of expression 25. What are the two methods of binary tree implementation? The two methods of binary tree implementation are: 1) Array or sequential representation 2) Linked list representation 26. What are the various operations of binary trees for link representation?

5 The various operations of binary trees for link representation are: 1) Creation of binary tree 2) Insertion of a node in the tree 3) Display of a binary tree 27. Define Pre-order Traversal To traverse a binary tree in Pre-order, the following operations are carried-out 1) Visit the root 2) Traverse the left subtree 3) Traverse the right subtree 28. Define Post-order Traversal To traverse a binary tree in Post-order, the following operations are carried-out 1) Traverse the left subtree 2) Traverse the right subtree 3) Visit the root 29. Define In-order Traversal To traverse a binary tree in In-order, the following operations are carried-out 1) Traverse the left subtree 2) Visit the root 3) Traverse the right sub tree 30. What is meant by binary search tree? [N/D 09][N/D 11] A binary search tree is also known as an ordered binary tree which is a node-based data structure in which each node has no more than two child nodes. Each child must either be a leaf node or the root of another binary search tree. The left sub-tree contains only nodes with keys less than the parent node and the right sub-tree contains only nodes with keys greater than the parent node. 31. Define Binary Search Tree Binary search tree is a special binary tree, which is either empty or should satisfy the following characteristics: 1) Every node has a value and no two nodes should have the same value i.e. the values in the binary search tree are distinct. 2) The value in any left sub-tree is less than the value of its parent node. 3) The value in any right sub-tree is greater than the value of its parent node. 4) The left and right sub-trees of each node are again binary search trees 32. Write an algorithm to find, insert and delete nodes in binary search tree.

6 Algorithm for insertion in Binary Search Tree: If key [x] < v move to the right child of x, else move to the left child of x When x is NIL, we found the correct position If v < key [y] insert the new node as y s left child else insert it as y s right child Algorithm for deletion in Binary Search Tree: if left[z] = NIL or right[z] = NIL then y z else y TREE-SUCCESSOR(z) if left[y] NIL then x left[y] else x right[y] if x NIL then p[x] p[y] if p[y] = NIL then root[t] x else if y = left[p[y]] then left[p[y]] x else right[p[y]] x if y z then key[z] key[y] copy y s satellite data into z return y 33. Define Graph [N/D 06] [N/D 07] A graph G consist of a nonempty set V which is a set of nodes of the graph, a set E which is the set of edges of the graph, and a mapping from the set for edge E to a set of pairs of elements of V. It can also be represented as G= (V, E). NONLINEAR DATA STRUCTURES Part B 1. Explain the concept of binary tree with example. A tree is a set of nodes which is either null or with one node designated as the root and the remaining nodes partitioned into smaller trees, called sub-trees. The level of a node is the length of the path from the root to that node, The depth of a tree is the maximum level of any node of any node in the tree

7 The degree of a node is the number of partitions in the subtree which has that node as the root Nodes with degree=0 are called leaves Binary tree is a tree in which the maximum degree of any node is 2. Binary tree is a finite set of elements that is either empty or is partitioned into three disjoint subsets. The first subset contains a single element called the root of the tree. The other two subsets are themselves binary trees called the left and right sub trees. The properties of binary tree the properties of binary tree are: 4) A binary tree of n elements has n-1 edges 5) A binary tree of height h has at least h and at most 2h - 1 elements 6) The height of a binary tree with n elements is at most n and at least log2 (n+1) Tree Traversal Algorithms: Traversing a tree means processing it in such a way, that each node is visited only once. Traversal is an operation which can be performed on a binary tree is visiting all the nodes exactly once. The different types of traversing are: 4) Pre-order traversal-yields prefix form of expression 5) In-order traversal-yields infix form of expression 6) Post-order traversal-yields postfix form of expression 1. In-order: 1. Traverse left subtree 2. Visit node (i.e. process node) 3. Traverse right subtree 2. Preorder: 1. Visit node 2. Traverse Left 3. Traverse right 3. Post-order: 1. Traverse left 2. Traverse right 3. Visit node Example: An arithmetic expression tree stores operands in leafs, operators in non-leaf nodes:

8 In-order traversal: (LNR) A-B+C/D*E-F Post-order traversal: (LRN) AB-C/EF-*+ Pre-order traversal: (NLR) +-AB*/CD-EF Applications: The applications of binary tree are: 4) Used in almost every high-bandwidth router for storing router-tables 5) Used in Huffman code construction 6) Used in parse tree construction 2. Explain the concept of binary search tree with example. A binary search tree is also known as an ordered binary tree which is a node-based data structure in which each node has no more than two child nodes. Each child must either be a leaf node or the root of another binary search tree. The left sub-tree contains only nodes with keys less than the parent node and the right sub-tree contains only nodes with keys greater than the parent node. Characteristics: Binary search tree is a special binary tree, which is either empty or should satisfy the following characteristics: 5) Every node has a value and no two nodes should have the same value i.e. the values in the binary search tree are distinct. 6) The value in any left sub-tree is less than the value of its parent node. 7) The value in any right sub-tree is greater than the value of its parent node. 8) The left and right sub-trees of each node are again binary search trees Operations in Binary Search Tree: Searching: Searching a binary search tree for a specific key can be a recursive or an iterative process. We begin by examining the root node. If the tree is null, the key we are searching for does not exist in the tree.

9 Otherwise, if the key equals that of the root, the search is successful and we return the node. If the key is less than that of the root, we search the left subtree. Similarly, if the key is greater than that of the root, we search the right subtree. This process is repeated until the key is found or the remaining subtree is null. If the searched key is not found before a null subtree is reached, then the item must not be present in the tree. This is easily expressed as a recursive Algorithm: function Find-recursive(key, node): // call initially with node = root if node = Null or node.key = key then return node else if key < node.key then return Find-recursive(key, node.left) else return Find-recursive(key, node.right) The same algorithm can be implemented iteratively: function Find(key, root): current-node := root while current-node is not Null do if current-node.key = key then return current-node else if key < current-node.key then current-node := current-node.left else current-node := current-node.right return Null Because in the worst case this algorithm must search from the root of the tree to the leaf farthest from the root, the search operation takes time proportional to the tree's height (see tree terminology). On average, binary search trees with n nodes have O(log n) height. However, in the worst case, binary search trees can have O(n) height, when the unbalanced tree resembles a linked list (degenerate tree). Insertion: Insertion begins as a search would begin; if the key is not equal to that of the root, we search the left or right subtrees as before. Eventually, we will reach an external node and add the new key-value pair (here encoded as a record 'newnode') as its right or left child, depending on the node's key. In other words, we examine the root and recursively insert the new node to the left subtree if its key is less than that of the root, or the right subtree if its key is greater than or equal to the root. Here's how a typical binary search tree insertion might be performed in a binary tree in C++: void insert(node*& root, int data) if (!root)

root = new Node(data); else if (data < root->data) insert(root->left, data); else if (data > root->data) insert(root->right, data); return; } To insert a new node in the tree, its key is first

10 root = new Node(data); else if (data < root->data) insert(root->left, data); else if (data > root->data) insert(root->right, data); return; } To insert a new node in the tree, its key is first compared with that of the root. If its key is less than the root's, it is then compared with the key of the root's left child. If its key is greater, it is compared with the root's right child. This process continues, until the new node is compared with a leaf node, and then it is added as this node's right or left child, depending on its key. There are other ways of inserting nodes into a binary tree, but this is the only way of inserting nodes at the leaves and at the same time preserving the BST structure. Deletion: Deletion of a node When we delete a node, we need to consider how we take care of the children of the deleted node. This has to be done such that the property of the search tree is maintained. Case 1: Node with no children If a node is a leaf node, it can be deleted immediately. Case 2: Node with one child It can be deleted by adjusting its parent pointer that points to its child node. Case 3: Node has 2 children Replace the data of the node to be deleted with its smallest data of the right subtree and recursively delete that node. Deleting a node with two children from a binary search tree. First the rightmost node in the left sub tree, the inorder predecessor 6, is identified. Its value is copied into the node being deleted. The inorder predecessor can then be easily deleted because it has at most one child. The same method works symmetrically using the inorder successor labeled 9. Example: Binary Search Tree for the input: 60, 25, 75, 15, 50, 66, 33,

11 Example: To delete To delete To delete

12 Algorithm to delete the nodes: searchtree deletion (int x, searchtree T) int t1; searchtree temp; if (T==NULL) printf ( Element not found ); else if (x < T->element) T->left = deletion (x, T->left); else if (x > T->element) T->right = deletion (x, T->right); else if (T->left && T->right) T1 = findmin (T->right); T->element = t1; T->right = deletion (T->element, T->right); } else temp = T; if (T->left = NULL) T=T->right; else if (T->right == NULL) T= T->left; free (temp); } } return T; 3. Explain the concept of graph traversal with example. A graph G consist of a nonempty set V which is a set of nodes of the graph, a set E which is the set of edges of the graph, and a mapping from the set for edge E to a set of pairs of elements of V. It can also be represented as G= (V, E). Depth-First Search: A depth-first search (DFS) in an undirected graph G is like wandering in a labyrinth with a string and a can of red paint without getting lost. We start at vertexs, tying the end of our string to the point and painting s visited. Next we label s as our current vertex called u.

13 Now we travel along an arbitrary edge (u,v). If edge (u,v) leads us to an already visited vertex v we return to u.if vertex v is unvisited, we unroll our string and move to v, paint v visited, set v as our current vertex, and repeat the previous steps. Eventually, we will get to a point where all incident edges on u lead to visited vertices. We then backtrack by unrolling our string to a previously visited vertex v. Then v becomes our current vertex and we repeat the previous steps. Then, if all incident edges on v lead to visited vertices, we backtrack as we did before. We continue to backtrack along the path we have traveled, finding and exploring unexplored edges, and repeating the procedure. When we backtrack to vertex s and there are no more unexplored edges incident on s, we have finished our DFS search. Algorithm DFS(v): Input: A vertex v in a graph Output: A labeling of the edges as discovery edges and backedges for each edge e incident on v do if edge e is unexplored then let w be the other endpoint of e if vertex w is unexplored then label e as a discovery edge recursively call DFS(w) else label e as a backedge Example: Depth-first traversal using an explicit stack. Sample Problem:

14 Order of Traversal A B C F E G D H I Stack The Preorder Depth First Tree: For a Graph G=(V, E) and n = V and m= E, 1) When Adjacency List is used, Complexity is O(m + n) 2) When Adjacency Matrix is used, Complexity is O (n 2 ). Because, Scanning each row for checking the connectivity of a Vertex is in order O(n). 4. Explain the breadth-first Search technique with example. Like DFS, a Breadth-First Search (BFS) traverses a connected component of a graph, and in doing so defines a spanning tree with several useful properties. The starting vertex s has level 0, and, as in DFS, defines that point as an anchor. In the first round, the string is unrolled the length of one edge, and all of the edges that are only one edge away from the anchor are visited. These edges are placed into level 1. In the second round, all the new edges that can be reached by unrolling the string 2 edges are visited and placed in level 2. This continues until every vertex has been assigned a level. The label of any vertex v corresponds to the length of the shortest path from s to v. Algorithm: void BFS(Edge G[], int vertex, boolean Visited[]) Edge tmp; int nextv; Queue Q = new Queue();

15 Q.enquque(new Integer(vertex)); while (!Q.empty()) nextv = ((Integer) Q.dequeue()).intValue(); if (!Visited[nextV]) Visited[next] = true; for (tmp = G[nextV]; tmp!= null; tmp = tmp.next) Q.enqueue(new Integer(tmp.neighbor())); }}} } Example: Breadth-first traversal using a queue. Sample Problem:

16 BFS Tree: (The BFS-tree for undirected graph is a free tree) A B D E C G F H I Algorithm Analysis: For a Graph G=(V, E) and n = V and m= E, 1) When Adjacency List is used, Complexity is O(m + n). 2) When Adjacency Matrix is used, Complexity is O(n 2 ). Because Scanning each row for checking the connectivity of a Vertex is in order O(n). Important Questions:. 1. Consider the following binary tree: G / \ D T / \ \ B M X / \ / \ \ A C I K Y \ P a) What is the result of a post-order traversal of the above tree? (8) b) What is the result of an in-order traversal of the above tree? (8) 2. Draw binary search tree for the following input list 60, 25, 75, 15, 50, 66, 33, 44}. Trace the algorithm to delete the nodes 25, 75, 44} from the tree. (16) 3. Explain in detail, the in-order, pre-order, and post-order traversal techniques. (8) [A/M 10] 4. Convert the expression ((A+B)*C-(D-E) ^ (F+G)) to equivalent prefix and postfix notations. (8) [A/M 10]

17 5. Explain in detail, the different methods of traversing a binary tree with algorithm. (8) [N/D 09] 6. What is binary search tree? Explain in detail, the various operations with an example. (16) 7. Explain in detail, the various types of graph traversals with neat algorithms. (16)

Partha Sarathi Mandal

MA 252: Data Structures and Algorithms Lecture 16 http://www.iitg.ernet.in/psm/indexing_ma252/y12/index.html Partha Sarathi Mandal Dept. of Mathematics, IIT Guwahati Deletion in BST Three cases Case 1: