Computational complexity of Fibonacci Sequence

Question

I understand Big-O notation, but I don\'t know how to calculate it for many functions. In particular, I\'ve been trying to figure out the computational complexity of the nai

Answer 1

The proof answers are good, but I always have to do a few iterations by hand to really convince myself. So I drew out a small calling tree on my whiteboard, and started counting the nodes. I split my counts out into total nodes, leaf nodes, and interior nodes. Here's what I got:

IN | OUT | TOT | LEAF | INT
 1 |   1 |   1 |   1  |   0
 2 |   1 |   1 |   1  |   0
 3 |   2 |   3 |   2  |   1
 4 |   3 |   5 |   3  |   2
 5 |   5 |   9 |   5  |   4
 6 |   8 |  15 |   8  |   7
 7 |  13 |  25 |  13  |  12
 8 |  21 |  41 |  21  |  20
 9 |  34 |  67 |  34  |  33
10 |  55 | 109 |  55  |  54

What immediately leaps out is that the number of leaf nodes is fib(n). What took a few more iterations to notice is that the number of interior nodes is fib(n) - 1. Therefore the total number of nodes is 2 * fib(n) - 1.

Since you drop the coefficients when classifying computational complexity, the final answer is θ(fib(n)).