What is the formula for finding various unmarked trees that can be formed from a given set of nodes?

Question

I am just doing a project research and ran into a problem. I would be very grateful if anyone could help me with this. Consider the following figure:

Two points connected by a line output only one diagram, three points connected by single lines also lead to the same figure no matter how you join the points, the result is the same. But as the points increase, there are different possibilities, as can be seen with four points.

Is there a formula for counting the number of unmarked trees that can be formed from a set of nodes?

Answer 1

.

n 2 ^ (ⁿ ₂) , ( ⁿ ₂) "n 2".

, , ( 100% , ).

. . MathWorld.

EDIT: n^(n-2).

Wikipedia.

Answer 2

, n . , ( n ^ {n-2}) ( 2 ^\binom {n} {2}).

- ( ): https://oeis.org/A000055. , A (x) , , ( ):

A (x) = 1 + T (x) - T ^ 2 (x)/2 + T (x ^ 2)/2, T (x) = x + x ^ 2 + 2x ^ 3 +...

, , . x ^ n n .

: http://austinmohr.com/work/trees. .