Uncover the logic behind the branching patterns of trees
By Marion Cromb
Have you ever noticed that on trees small twigs tend to stick out from thick branches at right angles, but branches of the same size split from each other at smaller angles? This is not a feature that is unique to arboreal branching, and in fact can be understood with a model for the human vascular system: Cecil D. Murray’s physiological principle of least work.
Work is the energy transported by a force, so this principle is about finding the configuration that expends the least energy. In other words; nature is lazy. Just as the arteries Murray studied transport blood throughout the body, we can model tree branches as a transportation network for water. Moving fluid along a narrow tube encounters larger frictional resistance (and thus takes more work) than moving fluid the same distance along a wide tube. So, to move from one point to another, taking a direct route in a small channel can be a lot less efficient than taking a longer, less direct route along a large channel then a short perpendicular hop in a narrow channel.
This principle of least work reveals a relationship in the angles between the channels and their diameters, which can be generalised to two cases. If two equal sized branches fork off the trunk on opposite sides, they do not deflect the trunk and emerge at the same angle. If just one branch emerges from the trunk, this will deflect the trunk, often considerably. Depending on the relative diameter of the trunk, branches come off at an angle between 70° and 90° to the original trunk, and the trunk is deflected between 0° and 90°.
To have a network that fills the available space efficiently (e.g. to collect the most sunlight), it is necessary to minimise the length of inefficient narrower channels (that can fill gaps between larger channels) whilst minimising the overall material used. This results in the branches that ‘feed’ the biggest areas being the thickest. Leonardo da Vinci observed that as a rule of thumb, the cross-sectional area of a branch is equal to the sum of the areas of the branches it splits into.
Of course, the trees themselves have not predetermined a high efficiency network to grow into, but grow in a modular fashion, obeying the same simple rules of cell division in the meristem at every stage of growth. But despite this fixed process trees don’t turn out as completely uniformly repeating structures because environmental factors come into play, for example competition for resources such as sunlight, and twigs snapping off in the wind. Those trees with growth rules that combine with these external factors to create efficient branching networks are those that evolution favours and that we see thriving today.
Branching is a result of very different mechanisms in many different physical phenomena: Lichtenberg figures, lightning and river systems to name a few. These branched networks all have similar properties and statistics to purely mathematical networks generated by random numbers, hinting that effective branching patterns are more dependent on the geometry of space itself than the processes behind them.
From Issue 14