(c) Evaluate this sum and then rewrite this as a function of n. You will probably have to [11 pts] look up an unfamiliar sum formula for this. Simplify this as much as possible - it will be very simple! Solution: 4. Suppose we have a perfect binary tree with height h 0 representing a heap, meaning it = has n 2+1 1 keys indexed from 1 to 2+1 1. When we run convertomaxheap we run maxheapify in reverse order on every key with children. Let's examine the worst-case - In the worst-case every single key gets swapped all the way to the leaf level. (a) For each level in the tree there are a certain number of nodes and each of those nodes [10 pts] requires a certain number of swaps. Fill in the appropriate values/expressions in the table: Level Number of Keys Number of Swaps per Key 0 2 .. (b) Write down a sum for the total number of swaps required. This should involve h, not n. [10 pts] Total

(c) Evaluate this sum and then rewrite this as a function of n. You will probably have to [11 pts] look up an unfamiliar sum formula for this. Simplify this as much as possible - it will be very simple! Solution: 4. Suppose we have a perfect binary tree with height h 0 representing a heap, meaning it = has n 2+1 1 keys indexed from 1 to 2+1 1. When we run convertomaxheap we run maxheapify in reverse order on every key with children. Let's examine the worst-case - In the worst-case every single key gets swapped all the way to the leaf level. (a) For each level in the tree there are a certain number of nodes and each of those nodes [10 pts] requires a certain number of swaps. Fill in the appropriate values/expressions in the table: Level Number of Keys Number of Swaps per Key 0 2 .. (b) Write down a sum for the total number of swaps required. This should involve h, not n. [10 pts] Total