Hi and welcome to Matladpi blog!

You may already know from my last year's posts how the language of the tree generation and enumeration triplet works, but I will recap the basics anyway. After all, we will be using the symbols of basic arithmetic so this differs a little from the original one.

Trees will be presented as sum forms that are like sums of numbers.

The first number 1 (one) in the sum form is the root vertex (the vertex with a label). It always starts the sum form.

The first number 1 inside a branch (highlighted with brackets) is an internal vertex. It always starts a branch and thus is always the first number inside the brackets.

The last number 1 that is not root or internal vertex, inside brackets and the whole sum form, is an external vertex. It is always the last number inside or outside the brackets.

+ is an edge and connects a number (branch or external vertex) to the root vertex (if the number is outside of brackets) or an internal vertex (if the number is inside the brackets this particular internal vertex starts).

( ) are brackets that are used to indicate that a branch is constructed. For example, the branch (1 + 1) = 2 is an internal vertex that is connected to an external vertex.

n (n > 2) is a branch called crossing where there is an internal vertex that is connected to n - 1 external vertices.

× is the multiplication sign that is used between a branch/ an external vertex and a quantity factor. For example, 1×2 means that there are two external vertices and 3×5 means that there are five crossings of size 3.

Examples:

1 is a lonely root vertex.

1 + 1 is the root vertex connected to an external vertex.

1 + (1 + 2×4 + 1×3) + 1×7 is the root vertex connected to 7 external vertices and a branch. In the branch there is the internal vertex that is connected to 3 external vertices and 4 branches 2 (internal vertex connected to an external vertex).

Notice that 2 = (1 + 1) and n = (1 + 1× (n - 1)). If there is no number 1 starting brackets, then these brackets do not construct a branch. Njäf! said.

I am Jesse Sakari Hyttinen and I will see you in the next post!