Trees are a perfect example. Fractal in form, down to a certain level; but also an example of something that builds over time like a cellular automaton and is being influenced by the environment. Compromises are made, the perfect form is impossible to achieve, but the gestalt is still there.
The fact that the most realistic simulated / computer-generated trees we can render are made primarily of simple fractals is a great indicator.
We see the same thing in terms of self-similarity in mountain ranges, lightning, rivers, lungs, and now clouds and water. At this point if you want to deny it, all I can surmise is that you either never go outside, or simply don't know what to look for.
I live in the mountains and I'm outside in a forest setting multiple times a week. I see exactly what you're talking about; it's just not the same as what the article is saying, even if you use the same word to describe it.