I was thinking about this recently since a post about drawing the Fibonacci sequence as trees caught my interest on r/math.

Essentially, they draw a point which bifurcates. Each of these two new points bifurcate, but 2 of the product are combined so that you get 3, then two of these three bifurcate while one produces 1, so you get 5. The product is a tree which goes 1, 2, 3, 5, etc. You can break this down into three types of points; "C"s, which are the product of a combination and therefore produce only one when they reproduce, "R"s, which are the product of a reproduction and therefore bifurcate, and "Twin R"s, which are two R's which share a common point parent and therefore two of their offspring combine. If you then draw it out with these names you see beautiful trees wherein branches of higher stages are reproductions of lower stages.

Begin with C, then

R

TR

R C R

TR R TR

RCR TR RCR

Etc.

The question is, why do these rules produce the Fibonacci sequence?

It seems to me that the Fibonacci sequence can be generated because n(2)(2)-n = n(2) n. E.g.,

1(2)=2

2(2)=4. Subtract 1, =3. Same as 1 2.

3(2)=6. Subtract 1, =5, same as 2 3.

5(2)=10. Subtract 2, =8. Same as 3 5.

8(2)=16. Subtract 3, =13. Same as 5 8.

13(2)=26. Substract 5, =21. Same as 8 13.

So what happens is that you have a doubling, then another doubling, then a substraction -- manifested as a combination -- portortional to the original n, then it repeats. It works because n(4)-n is the same as n(2) n, the reason each line is the sum of the previous two is due to this. Therefore you can draw the Fibonacci if you follow this rule.

Anyway, not sure this was coherent, but interesting to me to figure out why you can draw Fibonacci by rules of combination and bifurcation.