A structural visualization of the Collatz tree — Fibonacci branching pattern?
Hi everyone,
I’ve been exploring the inverse Collatz tree (predecessor tree) and noticed something interesting about its structure.
I built a modular tree based on residue classes modulo powers of 2. The tree is constructed layer by layer:
· Level 0: all integers (root)
· Level 1: odd/even split
· Level 2: residues mod 4
· Level 3: residues mod 8
· and so on...
What caught my attention is that the number of nodes at each level seems to follow the Fibonacci sequence:
Level 0: 1 node
Level 1: 2 nodes
Level 2: 3 nodes
Level 3: 5 nodes
Level 4: 8 nodes
Level 5: 13 nodes
Level 6: 21 nodes
Level 7: 34 nodes
I’ve attached two diagrams:
· 4-level tree (Nodes A–D)
· 8-level tree (Nodes A–H)
I’m curious:
· Has anyone else observed this Fibonacci branching in the Collatz tree before?
· Could this structural pattern be useful for understanding trajectory behavior?
I’m not claiming any proof here — just sharing a visual pattern I found interesting and would love to hear your thoughts.
Thanks for taking a look!
The preprint paper, complete with a full mathematical breakdown and written in LaTeX, is available for review on Zenodo at https://zenodo.org/records/21134642. The updated record provides direct access to the most recent version of the document.