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...