display | more...
Halcyon&on has a good point about fractal structure being a possible method of DNA conservation, but that is not the only possible reason for the occurance of fractal-like structures in living organisms. Fractal structures can also have advantages in other terms such as surface area to volume ratios. The branching structures in the alveoli of the lungs and the capillary beds of the circulatory system are good examples of such fractals. Their repeated branching structure gives them a high surface area to volume area, which is excellent for O2, CO2, and other chemical exchanges. The same can be said for the branching of trees, which uses small amounts of tissue to create a large area for leaves, which we all know absorb the light that the trees need.

More area = more light = more energy for the trees.
Less volume = less cells to support = less effort.

The fractal-like structure of these organisms is probably not directly coded in the DNA of the organism. There are lots of mathematical ways to generate fractals, but I doubt one would find any analogue to them in the DNA. It seems most likely that the complex interactions of systems of proteins are what result in these fractal-like structures, and that this being advantageous, was a property evolution decided to hang on to.

I say fractal-like to refer to such structures, because fractals are not rigidly defined in the mathematical literature, and also because these structures generally do not have all of the properties attributed to fractals, at least not in the mathematical sense. Mathematical fractals are self-similar on infinite scales, and are often predictably self-similar. Natural examples of fractals on the other hand, tend to have only qualitative self-similarity which can extend over large or small scales. It's a tricky thing to define, but the concepts of chaos theory and the fractal nature of real world objects are almost definitely linked.