Is there any exact formula for finding number of structurally different unlabeled trees can be formed with $n$ nodes? I searched a lot and I found some approximation formulas, e.g. here.
Check out A000055. Even if this is not what you want, it is probable that the OEIS has your sequence. The easiest way to look for a sequence in the OEIS is by calculating the first few numbers.
Apparently, no such formula is known. You can find a generating function, recursion, and first 1000 terms at OEIS.