1. Show that if f(n) is O(h(n)) and g(n) is O(i(n)), then f(n) + g(n) is O(h(n) + i(n)).
2. Show that 3(n + 1)7 + 2n log n is O(n7). Hint: Try applying the rules of Theorem 1.7. You will have to use the insert equations to answer this question.
3. Give an O(n)-time algorithm for computing the depth of each node of a tree T, where n is the number of nodes of T. Assume the existence of methods setDepth (v,d) and getDepth(v) that run in O(1) time.
4. What does the following algorithm do? Analyze its worst-case running time and express it using "Big-Oh" notation.
Algorithm Foo (a,n):
Input: two integers, a and n
k ß 0
b ß 1
while k < n do
k ßk + 1
b ß b *a
5. a. Describe (in pseudo-code) a findAll Elements (k) method of an AVL tree T. It should run in O(logn + s) time where n is the size of T and s is the number of elements returned (i.e., the number of nodes in T whose key is k).
b. Analyze the running time of your algorithm.
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