1. (‡) Prove asymptotic bounds for the following recursion relations. Tighter bounds will receive more marks. You may use the Master Theorem if it applies.

1. C(n) = 3C(n/2) + n

2. G(n) = G(n - 1) + 1/n

3. I(n) = I(n/2) + n/ lg(n)

2. Define a (p,q)-tree as a rooted tree where every internal node has between p and q (inclusive) children. Use the Master Theorem to give asymptotic bounds for the height of the tree. You can assume both p and q are constants with 2 ≤ p ≤ q.

3. (‡) Dominos

A 2 × 10 rectangle filled with ten dominos, and a 2 × 2 × 10 box filled with ten slabs.

1. A domino is a 2×1 or 1×2 rectangle. How many different ways are there to completely fill a 2 × n rectangle with n dominos?

2. A slab is a three-dimensional box with dimensions 1 × 2 × 2, 2 × 1 × 2, or 2 × 2 × 1. How many different ways are there to fill a 2 × 2 × n box with n slabs? Set up a recurrence relation and give reasonable exponential upper and lower bounds.