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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.

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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.

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  • $\begingroup$ See also A000081 for rooted trees (mentioned in FrankW's answer). $\endgroup$ – Yuval Filmus Sep 25 '14 at 14:48
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Apparently, no such formula is known. You can find a generating function, recursion, and first 1000 terms at OEIS.

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