Problem description:

You are climbing a staircase. It takes n steps to reach the top.

Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?

Example 1:

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Input: n = 2
Output: 2
Explanation: There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps

Example 2:

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Input: n = 3
Output: 3
Explanation: There are three ways to climb to the top.
1. 1 step + 1 step + 1 step
2. 1 step + 2 steps
3. 2 steps + 1 step

Constraints:

1 <= n <= 45

Solution:

Top-down

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class Solution:
    def climbStairs(self, n: int) -> int:
        if n == 1:
            return 1
        if n == 2:
            return 2
        return self.climbStairs(n-1)+self.climbStairs(n-2)

Bottom-up

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class Solution:
    def climbStairs(self, n: int) -> int:
        if n == 1:
            return 1
        dp = [0]* (n+1)
        dp[1], dp[2] = 1, 2
        
        for i in range(3, len(dp)):
            dp[i] = dp[i-1]+ dp[i-2]
        return dp[n]

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class Solution:
    def climbStairs(self, n: int) -> int:
        a = b = 1
        for _ in range(n):
            a, b = b, a+b
        return a

time complexity: $O()$
space complexity: $O()$
reference:
related problem: