I once suggested here (jokingly) the definition of mathematics as the study of SL_2(R) and related structures. With all due respect, taxonomy is not mathematics, even if you prefer a much broader definition. As I wrote some time ago, you are classifying ordered trees with respect to symmetry, which can be measured by the number of different labelling of the same tree.

To make a theory out of all this, you are welcome to pose and attempt to solve some real problems, such as:
- how fast does the number of those trees grows with the number n you are presenting? (does it get bigger than any polynomial in n? bigger than 2^n (2-to-the-power n)? is there a number c such that the number of trees, when divided by c^n, approaches 1?)
- what is the connection between the number of trees and the partition function p(), which counts the number of ways a number can be written additively (for example p(4)=5 since 4=3+1=2+2=2+1+1=1+1+1+1; this function was seriously studied first by Ramanujan).
- are there any natural operations that can defined on the set of trees, which will have interesting properties (such as x+y=y+x or x+(y+z)=(x+y)+z ) and transform this set into an object of some interest?
- can your notions of information and clarity degree be (first of all, defined, then) quantified beyond the level of yes-no? Can they be extended to infinite ordinals?

hi uzi

t(n is much bigger then p(n
And therefore any polynomial
And much more complicate so you cannot expect to have any simple formula

But there is written algorithm to calculate it

we are talking here
only about mathematics
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have bo different