0. 

1. examples:

def fib(n):

　　if n == 1 or n == 2:

　　　　result = 1

　　else 

   　　result = fib(n-1) + fib(n-2)

　　return result

this is very in-efficient, O(2^n), and we use the following solution, which is called memoized solution

def fib(n, memo):

　　if memo[n] != null:

　　　　return memo[n]

　　if n ==1 or n ==2:

　　　　result = 1

　　else 

　　　　result = fib(n-1, memo) + fib(n-2, memo)

　　memo[n] = result

　　return result

Memo is an array, store memo = {1, 1, 2, 3, 5, 8 ,...}

def fib_bottom_up(n):

　　if n ==1 or n ==2: 

　　　　return 1

　　bottom_up = new int[n+1]

　　bottom_up[1] =1

　　bottom_up[2] =1

　　for i from 3 to n:

　　　　bottom_up[i] = bottom_up[i-1] + bottom[i-2]

　　return bottom_up[n]