Sunset on a cold winter’s day. Look at the branching patterns on these trees – are they fractals? A fractal is anything that has self-similarity, meaning that you can zoom in or zoom out on the object without changing it – no scale is any more important than any other scale. Every part replicates every other part – all the way to infinitely large and small scales. A good example is the Mandelbrot set, a mathematical structure whose shape keeps reappearing the more one zooms in on it.
A little reflection will show that trees cannot be fractals in the strict sense, because they do not branch forever – there is a minimum twig size, and a maximum trunk size, no matter which tree you look at. But within this size range, are trees approximately fractals? Do the same branching rules apply for the small branches as for the large ones? This is a surprisingly hard question – to find out, you’d have to measure the sizes and positions of all the branches in a tree, and compare the branching rules for every level of branching. This can add up to thousands of branches very easily – imagine keeping track of every single one! And the problem gets worse, because most scientists would want to see replication (do the same results hold for multiple trees of the same species) and generality (does the result hold for multiple species). There are some researchers in my group who are working on testing these ideas right now, but they haven’t published their results yet – so stay tuned!
There is a weaker way to determine if trees are fractals – you can make a mathematical model that predicts several properties of trees (like how many leaves they should have, or how much carbon they use) based on their size and their branching rules. If these predictions are confirmed by observations, then it is reasonable to think trees might be approximately fractal. There is a large controversy over these measurements – they seem to work better for large trees than small ones, suggesting that small trees are not very fractal-like. Of course there is another option, too – another mathematical model (that has nothing to do with fractals) could make the exact same predictions. Until we can rule out all other theories, we are left without any definite conclusion. We will have to wait for the detailed measurements of all the branches in many species to be sure!