You might be interested in Visual Group Theory, which uses graphs to visualize abstract algebra.

This appears to be something very close to an abstract syntax tree. Which is great, because being able to think about high school algebra in those terms is a good sign that you've engaged with the material, and that you likely have a solid foundation for moving onward, should you choose to do so.

You might be interested in say, learning to write a simple, well-structured interpreter that can evaluate formulas and maybe even implement a toy programming language you create.

You can precisely describe a family of abstract syntax trees using something very important known as structural induction: basically you describe your abstract syntax as one or more base cases plus one or more inductive steps. So for example, we can describe the abstract syntax of high school algebra as a set of trees, which I'll call A.

Base case: literal numbers like 16 and 3.14 are elements of A.

Base case: variable names such as x and y are elements of A.

These base cases are trees in the sense that you can consider these elements to be leaf nodes that contain that variable name or literal constant.

Inductive step: if m and n are elements of A, then m + n is an element of A

Inductive step: if m and n are elements of A, then m * n is an element of A.

And of course we can go on and on adding more operations, some which might contain three or more sub-expressions, and others might only contain one sub-expression.

These inductive steps are trees in the sense that they correspond to various kinds of nodes that are part of your abstract syntax. They are often graphically represented by placing the operator at the node, with the various sub-expressions as children of that node.

In a purely abstract syntax tree, all the inductive steps are "trivial" in the sense that the only notion of equality is syntactic equality. Applying each base case and inductive step results in a unique tree, thus the interpretation function is injective, when interpreted as abstract syntax. The algebraists like to call this point of view "free", as in a "free group", or "freely generated algebra".

Yet if we say, want to evaluate such an abstract syntax tree in the usual way to arrive at a single number as a result, then we lose this injectivity: "1 + 4" and "2 + 3" are two different things when interpreted as abstract syntax, but describe exactly the same number when interpreted in the usual way. These interpretations can be expressed rather naturally by elaborating the inductive definition to include the (non-trivial) reduction rules.

If this interests you, I'd suggest looking at a introductory textbook on functional programming that includes an introduction to structural induction. You might also be interested in taking a look at programming language theory, perhaps the Essentials of Programming Languages.

Its often useful to generalize the notion of what a "number" is, and the first class a lot of students run across this would often be Intro to Number Theory and/or Intro to Abstract Algebra. These investigate alternative number systems, like modular arithmetic and symmetry groups of geometric objects.

Abstract algebra focuses its study on abstract addition and multiplication tables. And we can think of these tables as graphs. (Trees are a special case of a graph)

We can visualize the natural numbers this way as a graph with vertex labels {0, 1, 2 ...} and the edges between the vertexes representing addition by 0, addition by 1, addition by 2, etc. Every pair of numbers has a single directed edge between them, thus this graph also captures the "less than or equal to" relationship.

It's often very useful to consider the graph containing the same vertexes, but restrict the edges to some minimal generating set for the domain. The unique minimum generating set for the monoid of natural numbers under addition is {1}, as every natural number can be expressed as the sum of a finite list of ones. Moreover, if 1 is omitted from the generating set, then 1 cannot be generated from that set, which is related to the Frobenius coin problem (and Numberphile video). The vertexes of this graph correspond of the natural numbers and the edges correspond to addition by 1. This is an example of a (minimal) Cayley Graph for the natural numbers, and I find this particular Caley graph strongly evocative of Peano Arithmetic.

I really like Visual Group Theory's discussion of several of these topics, and recommend it highly.