Why is the complexity of computing the Fibonacci series 2^n and not n^2?

Question

I am trying to find complexity of Fibonacci series using a recursion tree and concluded height of tree = O(n) worst case, cost of each level = cn,

Answer 1

I cannot resist the temptation of connecting a linear time iterative algorithm for Fib to the exponential time recursive one: if one reads Jon Bentley's wonderful little book on "Writing Efficient Algorithms" I believe it is a simple case of "caching": whenever Fib(k) is calculated, store it in array FibCached[k]. Whenever Fib(j) is called, first check if it is cached in FibCached[j]; if yes, return the value; if not use recursion. (Look at the tree of calls now ...)