Let H,I,L, and N represent height,number of internal nodes,number of leaf nodes,and the total number of nodes respectively in a rooted binary tree.Which of the following statements is/are always TRUE?

Consider a hash table of size 10 with indices \( \{0, 1, \dots, 9\} \), with the hash function \[ h(x) = 3x \, (\text{mod} \, 10), \] where linear probing is used to handle collisions. The hash table is initially empty and then the following sequence of keys is inserted into the hash table: 1, 4, 5, 6, 14, 15. The indices where the keys 14 and 15 are stored are, respectively:

GATE DA - 2025
Data Structures

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Consider a directed graph \( G = (V,E) \), where \( V = \{0,1,2,\dots,100\} \) and \[ E = \{(i,j) : 0<j - i \leq 2, \text{ for all } i,j \in V \}. \] Suppose the adjacency list of each vertex is in \textit{decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0. The number of vertices that will be discovered after vertex 50 is:}

GATE DA - 2025
Data Structures

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Suppose that insertion sort is applied to the array \( [1, 3, 5, 7, 9, 11, x, 15, 13] \) and it takes exactly two swaps to sort the array. Select all possible values of \( x \).

GATE DA - 2025
Searching, Sorting and Hashing

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Consider the following pseudocode.

Create empty stack S
Set x = 0, flag = 0, sum = 0
Push x onto S
while (S is not empty){
    if (flag equals 0){
        Set x = x + 1
        Push x onto S
    }
    if (x equals 8):
        Set flag = 1
    if (flag equals 1){
        x = Pop(S)
        if (x is odd):
            Pop(S)
        Set sum = sum + x
    }
}
Output sum

The value of \( sum \) output by a program executing the above pseudocode is: