Solution

The correct answer is A is true but R is false

EXPLANATION:

• Assertion A: "AVL trees are more balanced as compared to Red Black trees, but they may cause more rotations during insertion and deletion."
  • This assertion is correct. AVL trees are more strictly balanced than Red-Black trees, meaning that the height difference between the left and right subtrees of any node (called the balance factor) is at most 1 in AVL trees, whereas it can be up to 2 in Red-Black trees.
  • However, this strict balancing in AVL trees can lead to more rotations during insertion and deletion operations compared to Red-Black trees.
• Reason R: "A Red-Black tree with n nodes has a height that is greater than 2 log₂ (n + 1), and the AVL tree with n nodes has a height less than logΦ (√5 (n+2)) - 2 (where Φ is the golden ratio)."
  • This reason is incorrect. The correct upper bound for the height of a Red-Black tree with n nodes is 2 log₂ (n + 1), not greater than.
  • The statement for AVL trees is also incorrect; the correct upper bound for the height of an AVL tree with n nodes is approximately 1.44 * log₂(n+2) - 1.329.

Therefore, Assertion A is correct, but Reason R is incorrect. The correct answer is option 3: "A is true but R is false."