Finding the time complexity of fibonacci sequence

Question

I tried it as follows and would like to know if it is correct.

Answer 1

The analysis is not accurate although the result is correct. You could write it more accurately by replacing $=$ with $\le$

$T(n) \le c(1+2+..+2^{n-1})$ ( $\le$ since not all level have same number of children, consider the most right-handed path, n is decreasing by $2$ every step ).

Indeed a more careful analysis can get you a tighter bound as mentioned in the comment. The idea is, the time $T(n)$ is computed with $T(n-1) + T(n-2)$ the same way as the actual fibonacci $F(n)$, and since $F(n) = O(\phi^n)$ for $\phi = (1+\sqrt{5})/2$ as the closed form.

Thus $T(n) = O(\phi^n)$ which is slightly smaller than $2^n$