> where X is a random topic which is mostly or completely unrelated to Tree Notation)

What is binary notation related to? Is it related to medicine? It is related to automobiles? Is it related to deep space? Is it related to food? Is it related to government? Try to do any of those things without binary notation. This is a big idea. A very infrequent big idea.

So what are your definitions of tree and triangle?

Mathematics is not a rigorous language, so I'll put the crux of my argument in Tree Notation, which is a rigorous language:

    betterGraphTheoryLanguage
     nodeCell
     cycleNode
     triangleNode
      extends cycleNode
      cells nodeCell nodeCell nodeCell

typo fix: changed Trees to Tree Notation.

Uh huh. Great start...

> I'll put the crux of my argument in Trees, which is a rigorous language

Interesting how you're overloading the word "tree" in this context. You're presenting a definition in the format of a tree. And concluding that the resulting object is itself a tree. It's like saying that swords are made of wood, and proving it by carving the definition of a sword into a piece of wood. This is quackery.

Graphs, and trees, have perfectly rigorous definitions. They've been defined in a plurality of rigorous languages. Your "definition" doesn't even mention edges, unless they're implied by "cycleNode" somehow; in either case it appears that your "better" graph theory only contains a single graph of any given order. Graph theorists expect a few more than that.

Let me define a "rigorous" language.

A rigorous langauge is one which can be defined completely in a single document starting with 0 and 1 and working it's way all the way up to whatever it is you want to prove.

> Graphs, and trees, have perfectly rigorous definitions. They've been defined in a plurality of rigorous languages.

If you can provide a link to a single page that starts with 0 and 1 and builds all the way up to defining "graphs and trees" and how a "triangle is not a tree", then I will agree with you.

> Interesting how you're overloading the word "tree" in this context.

Sorry, that was a typo, I meant to write Tree Notation. I'll fix.

Is there a document like this for your tree notation? Not trying to criticize; I'm legitimately interested to see how this would be done.

I wish. We have a lot of ongoing threads about it. Someone has one somewhere for logic gates, IIRC.

Here's a snippet from 1 from a conversation the other day on how to build up to an if statement.

  0
  1
  2 10
  ...
  i 01101001
  ...
  f 01100110
  ...
  if 1
   2

The idea is we'd like to be able to compare 2 systems objectively and decide which is the simpler one.

Here's a rough paper from 2017 (before I had any experience in academia): https://github.com/treenotation/jtree/blob/master/papers/pap...