Count the Ones

Alice recently converted all the positive integers from $1$ to $2^N - 1$ (both inclusive) into binary strings and stored them in an array $S$. Note that the binary strings **do not have leading zeroes**. While she was out, Bob **sorted** all the elements of $S$ in lexicographical increasing order. Let $S_i$ denotes the $i^{th}$ string in the **sorted array**. Alice defined a function $F$ suc