Hi and welcome to the Matladpi blog!

The theorem that I have discovered quite a while ago about mathematical trees has some pretty interesting consequences. For a remainder, the theorem is stated as the following:

'Any nonempty and finite tree can be expressed as a partition of a positive whole number. '

Note that the term 'tree' means a tree without any labels for its edges, vertices et cetera, with the exception of a rooted tree that has a labeled vertex known as a root. But, the theorem can be expanded in a way that we create a tree language, which makes it possible to include labels as well.

The theorem in its simplest form is used mainly though for free and rooted trees, and series reduced free and rooted trees as can be witnessed in the case of my book, which can be found in this site under the menu heading 'Books of Jesse Sakari Hyttinen'.

What comes to the consequences, as already mentioned, the tree language can be formed. In its simplest form, also known as partition form, we can see the deep connection between rooted tree listing and partition listing, and as a clear consequence, the enumeration cases as well. This is the second consequence.

For example, if we want to list the partitions of a number 5, we could write

1*5 = 2+1*3 = 3+1*2 = 4+1 = 5

= 2*2+1 = 3+2

So in total, there are 7 partitions.

Notice now how the case is when we want to list all the so-called 'basic forms' of rooted trees with tree size 6:

6 = 1+1*5 = 1+2+1*3 = 1+3+1*2

= 1+2*2+1 = 1+3+2

= 1+4+1 = 1+5

Notice how here there are 7 forms (the number 6 is not a rooted tree, but the whole forest whose trees we are listing) as well? And that the forms look like each other, with the exception of the number 1 added to the trees case? This is no random event, but the consequence of the tree theorem.

Now you may ask what a 'basic form' is. Well, in the case of this language, it means all those trees which can be expressed as partitions without parentheses. If we include the advanced forms as well, the result would be this:

6 =1+1*5 =1+2+1*3 = 1+3+1*2

=1+2*2+1 = 1+(1+2)+1*2

= 1+3+2

= 1+(1+2)+2

= 1+4+1

= 1+(1+2+1)+1

= 1+(1+3)+1

= 1+(1+(1+2))+1

= 1+5

= 1+(1+2+1*2)

= 1+(1+2*2)

= 1+(1+3+1)

= 1+(1+(1+2)+1)

= 1+(1+4)

= 1+(1+(1+2+1))

= 1+(1+(1+3))

= 1+(1+(1+(1+2)))

So, there are 20 rooted trees with tree size 6.

The advanced forms and basic forms of trees together form the forest, and the tree listing examples here have an order that is provided by my tree listing/ generation and enumeration algorithms. These algorithms with the language and theorem, and also the condition matrix, form the tree generation and enumeration triplet, the topic of my book I have in this site.

I am Jesse Sakari Hyttinen and I will see you in the following post, which is all about constructing and expressing trees as partitions.