When I look at this function, I have no idea what the time complexity is for it. How do you guys go on about finding the time complexity of a recursive function?

From that recursive function, the complexity is probably just 2^n, because each recursive call leads to 2 more recursive calls, and so on and so forth.

Yes, because the outer loop iterates 2^N times and the inner loop iterates N times. We need to run an inner loop in O(N) to get all the indexes of the “on” bits of b and add them to the subset.

For the recursive way, I like to mentally think of a tree of recursive calls. In total, k will range from 0 to N inclusive in all the calls to search. k is the level of the call/node in the tree, so the total depth of the tree is N. There is only 1 call at the first level of the tree, but the number of calls/nodes at a specific level doubles each time we increase the level by 1, because each recursive call leads to 2 more recursive calls (as said by SansPapyrus683).

Thus, we can generate an upper bound of the complexity to be O(N \cdot 2^N), as there are roughly N levels and each level has an upper bound of 2^N nodes.

Of course, for both implementations, O(N \cdot 2^N) can be simplified to just O(2^N).

I personally have never encountered a problem where the time is so tight that the difference of a factor of 10 in the time complexity would have been the difference between the number of operations being too low or too high.

I myself have encountered problems where the difference between TLE and AC was a difference in the constant factor as minor as switching from an ArrayList to an array.