What is the number of binary trees with 3 nodes which when traversed in postorder give the sequence A, B, C? Draw all these binary trees.

The preorder traversal of a binary search tree is 15, 10, 12, 11, 20, 18, 16, 19.

Which one of the following is the postorder traversal of the tree?

Consider a double hashing scheme in which the primary hash function is h₁(k) = k mod 23, and the secondary hash function is h₂(k) = 1 + (k mod 19). Assume that the table size is 23. Then the address returned by probe 1 in the probe sequence (assume that the probe sequence begins at probe 0) for key value k=90 is _______.

What is the worst case time complexity of inserting n elements into an empty linked list, if the linked list needs to be maintained in sorted order?

Consider the following C program.

#include 
int main ()  {
     int a [4] [5] = {{1, 2, 3, 4, 5},
                      {6, 7, 8, 9, 10},
                      {11, 12, 13, 14, 15},
                      {16, 17, 18, 19, 20}};
     printf (“%d\n”, *(*(a+**a+2) +3));
     return (0);
} 

The output of the program is _______.

What is the worst case time complexity of inserting n² elements into an AVL-tree with n elements initially?

Let G = (V,E) be a directed, weighted graph with weight function w:E → R. For some function f:V → R, for each edge (u,v) ∈ E, define w'(u,v) as w(u,v) + f(u) - f(v).
Which one of the options completes the following sentence so that it is TRUE?

“The shortest paths in G under w are shortest paths under w’ too, _______”.

In a balanced binary search tree with n elements, what is the worst case time complexity of reporting all elements in range [a,b]? Assume that the number of reported elements is k.

Consider the array representation of a binary min-heap containing 1023 elements. The minimum number of comparisons required to find the maximum in the heap is _______.

Consider the following statements:

(i) First-in-first out types of computations are efficiently supported by STACKS.
(ii) Implementing LISTS on linked lists is more efficient than implementing LISTS on an array for almost all the basic LIST operations.
(iii) Implementing QUEUES on a circular array is more efficient than implementing QUEUES on a linear array with two indices.
(iv) Last-in-first-out type of computations are efficiently supported by QUEUES.

Which of the following is correct?

An advantage of chained hash table (external hashing) over the open addressing scheme is

In the balanced binary tree in the below figure, how many nodes will become unbalanced when a node is inserted as a child of the node “g”?

Which of the following sequences denotes the post order traversal sequence of the tree of question 14?

The minimum number of interchanges needed to convert the array

 89, 19, 40, 17, 12, 10, 2, 5, 7, 11, 6, 9, 70  

into a heap with the maximum element at the root is

A binary search tree is generated by inserting in order the following integers:

 50, 15, 62, 5, 20, 58, 91, 3, 8, 37, 60, 24  

The number of nodes in the left subtree and right subtree of the root respectively is

A binary search tree is used to locate the number 43. Which of the following probe sequences are possible and which are not? Explain.

The concatenation of two lists is to be performed in O(1) time. Which of the following implementations of a list should be used?

Which of the following is essential for converting an infix expression to the postfix from efficiently?

A binary search tree contains the value 1, 2, 3, 4, 5, 6, 7, 8. The tree is traversed in pre-order and the values are printed out. Which of the following sequences is a valid output?

A priority queue Q is used to implement a stack that stores characters. PUSH (C) is implemented INSERT (Q, C, K) where K is an appropriate integer key chosen by the implementation. POP is implemented as DELETEMIN(Q). For a sequence of operations, the keys chosen are in

The postorder traversal of a binary tree is 8, 9, 6, 7, 4, 5, 2, 3, 1. The inorder traversal of the same tree is 8, 6, 9, 4, 7, 2, 5, 1, 3. The height of a tree is the length of the longest path from the root to any leaf. The height of the binary tree above is _______.

A queue is implemented using a non-circular singly linked list. The queue has a head pointer and a tail pointer, as shown in the figure. Let n denote the number of nodes in the queue. Let 'enqueue' be implemented by inserting a new node at the head, and 'dequeue' be implemented by deletion of a node from the tail.

Which one of the following is the time complexity of the most time-efficient implementation of 'enqueue' and 'dequeue, respectively, for this data structure?

The number of possible min-heaps containing each value from {1, 2, 3, 4, 5, 6, 7} exactly once is ___________.

Which of the following statements is false?

A complete n-ary tree is one in which every node has 0 or n sons. If x is the number of internal nodes of a complete n-ary tree, the number of leaves in it is given by

Let A be a two dimensional array declared as follows:

  A: array [1 ... 10] [1 ... 15] of integer;  

Assuming that each integer takes one memory location, the array is stored in row-major order and the first element of the array is stored at location 100, what is the address of the element a[i][j]?

Faster access to non-local variables is achieved using an array of pointers to activation records called a

Let p be a pointer as shown in the figure in a single linked list.

What do the following assignment statements achieve?

q: = p → next
p → next:= q → next
q → next:=(q → next) → next
(p → next) → next:= q

(b) Compute the postfix equivalent of the following Infix expression

  3 * log(x+1) - a/2  

Draw the binary tree with node labels a, b, c, d, e, f and g for which the inorder and postorder traversals result in the following sequences:

 Inorder       a f b c d g e
 Postorder     a f c g e d b 

(a) Derive a recurrence relation of the size of the smallest AVL tree with height h.

The number of articulation points of the following graph is

(a) In a binary tree, a nil node is defined to be a node with 2 children. Use induction on the height of the binary tree to prove that the number of full nodes plus one is equal to the number of leaves.

(b) Draw a min-heap that results from insertion of the following elements in order into an initially empty min-heap: 7, 6, 5, 4, 2, 3, 1. Show the result after the deletion of the root of this heap.

The most appropriate matching for the following pairs

          X: depth first search            1: heap
          Y: breadth-first search          2: queue
          Z: sorting                       3: stack  

is

Consider the following nested representation of binary trees: (X Y Z) indicates Y and Z are the left and right sub stress, respectively, of node X. Note that Y and Z may be NULL, or further nested. Which of the following represents a valid binary tree?

Suppose you are given an array s[1...n] and a procedure reverse (s,i,j) which reverses the order of elements in a between positions i and j (both inclusive). What does the following sequence do, where 1 ≤ k ≤ n:

         reverse(s, 1, k) ;
         reverse(s, k + 1, n);
         reverse(s, l, n);  

Let LASTPOST, LASTIN and LASTPRE denote the last vertex visited in a postorder, inorder and preorder traversal. Respectively, of a complete binary tree. Which of the following is always tree?

Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-first search tree. Let u be a vertex in G and let ν be the first new (unvisited) vertex visited after visiting u in the traversal. Which of the following statements is always true?

Suppose a stack implementation supports, in addition to PUSH and POP, an operation REVERSE, which reverses the order of the elements on the stack.
(a) To implement a queue using the above stack implementation, show how to implement ENQUEUE using a single operation and DEQUEUE using a sequence of 3 operations.
(b) The following postfix expression, containing single digit operands and arithmetic operators + and *, is evaluated using a stack.

 5 2 * 3 4 + 5 2 * * + 

Show the contents of the stack.

(i) After evaluating 5 2 * 3 4 +
(ii) After evaluating 5 2 * 3 4 + 5 2
(iii) At the end of evaluation.

Suppose you are given arrays p[1…..N] and q[1…….N] both uninitialized that is, each location may contain an arbitrary value), and a variable count, initialized to 0. Consider the following procedures set and iset:

 set (i) {
      count = count + 1;
      q [count] = i;
      p[i] = count;
 }
 is_set(i) {
      if (p[i] ≤ 0 or p[i] > count)
      return false;
      if (q[p[i]] ≠i)
      return false;
 return true;
 } 

(a) Suppose we make the following sequence of calls: set (7); set (3); set(9);
After these quence of calls, what is the value of count, and what do q[1], q[2], q[3], p[7], p[3] and p[9] contain?

(b) Complete the following statement “The first count elements of _______ contain values i such that set (__________) has been called”.

(c) Show that if set (i) has not been called for some i, then regardless of what p[i] contains, is_set (i) will return false.

Consider any array representation of an n element binary heap where the elements are stored from index 1 to index n of the array. For the element stored at index i of the array (i≤n), the index of the parent is

Consider an undirected unweighted graph G. Let a breadth-first traversal of G be done starting from a node r. Let d(r,u) and d(r,v) be the lengths of the shortest paths from r to u and v respectively in G. If u is visited before v during the breadth-first traversal, which of the following statements is correct?

What is the minimum number of stacks of size n required to implement a queue of size n?

Consider the following three C functions:

[PI]            int*g(void) 
             { 
                int x = 10; 
                return(&x); 
             }  
    
[P2]            int*g(void) 
             { 
                int*px; 
                *px = 10; 
                return px; 
             } 
    
[P3]            int*g(void) 
             { 
                int*px; 
                px = (int *) malloc(sizeof(int)); 
                *px = 10; 
                return px; 
             } 

Which of the above three functions are likely to cause problems with pointers?

(a) Insert the following keys one by one into a binary search tree in the order specified.

         15, 32, 20, 9, 3, 25, 12, 1  

Show the final binary search tree after the insertions.
(b) Draw the binary search tree after deleting 15 from it.
(c) Complete the statements S1, S2 and S3 in the following function so that the function computes the depth of a binary rooted at t.

              typedef struct tnode{
                  int key;
                  struct tnode *left, *right;
              } *Tree;
              int depth(Tree t)
              {
                  int x,y;
                  it (t == NULL) return0;
                  x=depth(t → left);
 S1:              ____________;
 S2:              if(x>y) return _____________:
 S3:              else return _____________;
              } 

The results returned by function under value-result and reference parameter passing conventions