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?
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