TITLE: Göbel's correspondance between rooted trees and natural numbers
QUESTION [10 upvotes]: In the paper On a 1-1-correspondence between rooted trees and natural numbers by F. Goebel, a correspondence between natural numbers and rooted tree was established via prime factorization.
He defines:

Let $T$ be a rooted tree, $r$ its root. The connected components of $T-r$ are denoted by $T_1,\dots,T_v$, where $v$ is the degree of $r$. The graphs $T_j$ ($j=1,\dots,v$) obviously are trees, which we transform into rooted trees by defining as the root of $T_j$ the vertex of $T_v$ which is adjacent to $r$ in $T$.

Figure 2 (below) shows all rooted trees up to n=45. I do understand all composite numbers, however I fail to understand the structure of how the rooted trees for prime numbers are structured.

Question: What are the rules to get the rooted tree for prime numbers?

REPLY [14 votes]: (I see this in the comments, too, but to ensure this has an actual answer...)
Here are the first fifteen natural numbers after drawing an individual line segment (edge and node) beneath the root:

As you can see, I also numbered what this change does:
The $n$th natural becomes the diagram for the $n$th prime, e.g., you recover the rooted trees beginning with $2, 3, 5, 7, 11, 13$ by looking at this image. Next will be $17, 19, 23$, etcetera.