Example 4 Perform the following operations on a positive integer: if it is even, divide it by 2; if it is odd, add 1. Continue doing this until the number becomes 1 and the operation stops. Find how many numbers will become 1 after 9 operations.
The number that becomes 1 after 1 operation is only 1, which is 2;

The number that becomes 1 after 2 operations is also only 1, which is 4;
The numbers that become 1 after 3 operations are 2, which are 3, 8; ......
It can be observed that if a number $a$ is odd, there is only 1 way to get $a$; if it is even, there are 2 ways to get $a$. Therefore, the number of ways to get 1 after $n$ operations satisfies the Fibonacci sequence's recurrence relation. The answer is $F_{n}$.