ISRO CSE 2023

Detailed Explanation:

$Vertices n = 5 (A, B, C, D, E). Minimum Spanning Tree must contain n - 1 = 4 edges.$

Select smallest edges avoiding cycles:
- AB = 1
- DE = 2
- BE = 3
- BC = 5

Total MST weight = 1 + 2 + 3 + 5 = 11.

Number of edges = 4.

Detailed Explanation:

Handshaking Lemma states sum of degrees equals twice the number of edges.

An Euler graph is a connected graph where every vertex has even degree.

Planar graphs can be drawn without edge crossings.

Chromatic number is min colors needed for vertex coloring.

Detailed Explanation:

The diameter of a binary tree is the length of the longest path between any two nodes.

By calculating the height and diameter recursively in a bottom-up post-order traversal, each node is visited once.

Detailed Explanation:

Preorder: Root=10. Left subtree values < 10: (5, 1, 7). Right subtree values > 10: (40, 50).

Left subtree root=5, left=1, right=7 -> Postorder: 1, 7, 5.

Right subtree root=40, right=50 -> Postorder: 50, 40.

Detailed Explanation:

Rewrite using Horner's rule: f(z) = z * (8 * z * z + 3) + 12.

Operations: 3 multiplications (z*z, 8*($z^2)$, z*(...)) and 2 additions (+3, +12).