Here's a collatz tree that i drew.
It contains the powers of two along its backbone, and $(2^n-1)/3$ where possible. Those numbers themselves have each a backbone of their own with $(2^nk-1)/3$ where possible. Note that odd numbers do not have branches since we do not apply $3n+1$ to even numbers. The backbones highlighted in green are 'dead'. They are divisible by 3 so no vertex can have an integer $(2^nk-1)/3$. (I made a mistake though, 53 is not meant to be highlighted).
The collatz conjecture claims that every number is contained within this tree and that the only cycled or repeated number is $1$ which is labelled R in a circle and is branching from $2^2$. Anyway, my question is how do you explain or express the regularity of this 'fractal'? This includes the fact that the 'multiplier' leading to branches at every backbone is either two or four and each new branch is separated by $2^2$. Also is there a regularity in where the green backbones pop up? I imagine it can be shown through modular arithmetic.
It was a pain drawing this. I am open to any suggestions on computational methods, i.e. a fractal generator or something in order to make it look clearer.
In a compressed format (omitting the even numbers) an approach to draw that tree (in Excel) is this one:
The blocks of numbers are meant as infinite to the right side. The numbers, read downwards in the columns and then horizontal across the columns are in a sequence $a_{k+1}=4 a_k+1$ and the colored arrows show the dependance of child-boxes to parent-entries. For instance, the red arrow from the $17$-box to its parent $13$ means $17=(13 \cdot 4-1)/3$ where the multiplier is $4$ and the green arrow from the box $227$ to its parent $341$ means $227 = (341 \cdot 2-1)/3$ where the multiplier is $2$.
It can surely be optimized, though..., perhaps I come back to this, later.