Couldn’t get this idea outta my head, whether it’s reasonable or not…

Happy Thanksgiving!
— 

Given some binary tree (doesn’t have to be just a binary tree though, I guess):

 
 

 
 
A
 /       \ 
B        C
 /    \     /    \
D   E    F   G
 
 
It seems possible to represent the tree with a bunch of ordered n-tuples and relations.

First, let A = (A, A). In this context, "=" really just stands for "R" (relation). And it’s a little pedantic to say that A R {A and A}, but for me it made it clear that A=(A, A) means A R A and A R A.

With that being said, let B = (D, E) and let C = (F, G).

A is related to B as well as C, written as A = B and A = C, which could be written equivalently as (A, A)=(B, C); A R {B and C}.

But since A = (A, A) = (B, C) and B=(D, E); C=(F,G), its also true that–
 

 
 
(B, C) = ((D, E), (F, G)).
 
So, A = (A, A) = (B, C) = ((D, E), (F, G)). (1)

A is related to itself and is related to B and C too. And,  B is related to (D, E), and C is related to (F, G).. 

Breaking (1) down further:

A = (A, A): A R A ^ A = A R A = (A, A) (derp);
A = (B, C): A = A R B ^ C = (A R B and A R C) = {(A, B), (A, C)};
B = (D, E): (B, C) = B R D ^ E = (B R D and B R E) = {(B, D), (B, E)};
C = (F, G): C R F ^ G = C R F and C R G = {(C, F), (C, G)}.

Lol, I don’t know if I got some terminology right (notably relations), but its nice to know that if we look at the levels of the binary tree, taking the cartesian product of lvl1 x lvl2, then lvl2 x lvl3, we have:
 

 
lvl1: {A}
lvl2: {B, C}
lvl3: {D, E, F, G}.

Then, taking the Cartesian product of the levels:

lvl1 x lvl2 = {A} x {B, C} = {(A, B), (A, C)}
lvl2 x lvl3 = {B, C} x {D, E, F, G} = {(B, D), (B, E), (B, F), (B, G), (C, D), (C, E), (C, F), (C, G)}.
 

I guess I’m not completely off, but the whole A = (A, A) thing is still silly and may lead to trouble though. Seems that way anyways.