Fractals is a branch of mathematics that artists intuitively grasp because it appeals strongly to the pattern recognition part of the right brain. One of the principles of fractals is the concept of self-similarity. Many forms in nature repeat a certain structure or geometry at various levels of scale.
For example, this ginger root has bulbous branches that leave the main stem, and they tend to do the same thing as the branches get smaller, until the pattern degenerates at the smallest scale.
Romanesco broccoli demonstrates the principle of self-similarity even better. Each spiraling cone is composed of smaller spiraling cones. (Click to enlarge).
In many ferns, the shape of the leaflet resembles the frond. Since this comes down to math, computers can easily invent self-similarity, and this component has given realism to CGI renderings of rocks, plants, clouds, and water, where the phenomenon appears everywhere. Check out the digital fern below, which "grows" as you scroll down.
Fractal fern, link.
Wikipedia on fractals, link.
Fern glossary and more photos at: link.