Recently I was reading the book Scale by the theoretical physicist Geoffrey West. Much of the book is devoted how scaling relationships control the behavior of various phenomena, especially in the biological and socioeconomic realm.
Even prior to reading this book it was to my understanding that the underlying signature of behavior for self-similarity was that the process obeyed a power law, that is, that it is scale invariant. No matter what level of zoom or resolution we choose, the function or shape will look the same.
West gives an example of arterial branching which he says is similar to Leonardo da Vinci's realization that trees display an area-preserving relationship, wherein at every branching the daughter branches' cross-sectional area add up to the cross-sectional area of the parent. For instance, at the first branching each daughter branch has $1/2$ the area of the parent. They sum to $1$, the area of the parent. At the 2nd branching there are now $4$ daughter branches, each $1/4$ the area of the original parent. At every level of branching the $2$ branches sum to the branch above them, as well.
West makes note that the radii of successive branches decrease by a factor of $\sqrt{2}$. After, say, $10$ branchings, the radius of the 10th artery/branch would be $1/32$ the radius of the original branch. He calls this a fractal relationship. How is this a fractal and how does it display self-similarity when it is exponential behavior and not a power law? What we are seeing is $\sqrt{2}^N$. The exponent is the variable, not the base, as in a power law.
Furthermore, mathematical constructions like the Sierpinksi Triangle or the Fractal Snowflake are considered fractals, but are they actually self-similar? The process of subdivision is exponential for these objects, with $3^n$ being the case for Sierpinski Triangle.
What am I either missing, misunderstanding, or confusing here? Can exponential functions exhibit "fractal" behavior but not be self-similar? I understand there are subtle differences between scale-invariance and self-similarity but could someone elaborate? This has been really bugging me.