## Friday, April 14, 2017

### Fibonacci sequence and the worst AVL trees

Fibonocci numbers: We all know about Fibonacci sequence. It is the famous sequence that starts with 1 and 1 as the first two elements. From there on, every element is the sum of the previous elements. So, it goes like 1,1,2,3,5,8,13,21,... We can call the $n^{th}$ element this series $F_n$. So, we have $F_1=1$, $F_2=1$ and $F_n = F_{n-1} + F_{n-2}$. Fibonooci numbers are very closely related to AVL trees, a kind of self-balancing binary search tree. We will explore this relation in this blog article and then establish the worst case search complexity of an AVL tree.

A formula for Fibonacci numbers: Our problem is to find out any arbitrary element $F_{n}$ given $n$. Sure, one can just start from the beginning and keep computing each element until $F_n$ is reached, but this probably is not the most efficient method of computing $F_n$. It will be nice to have a formula for $F_n$. This is what we are going to find out.