Da Vinci's rule of trees doesn't hold water

Image: Arnaud Mesureur / Unsplash

The 'rule of trees' developed by Leonardo da Vinci to describe how best to draw them, has been disproved by scientists applying the theory to the vascular systems of trees.

The Italian Renaissance polymath, who was born in 1452, was a painter, draughtsman, engineer, scientist, theorist, sculptor and architect. His interest in drawing led him to look at size ratios of objects, including trees, so that he could create more accurate representations.

He perceived a rule of trees, which states that "all the branches of a tree at every stage of its height are equal in thickness to the trunk when put together."

It had been thought that the rule of trees could also be applied to the vascular channels which transport water and nutrients through a tree, with the individual channel sizes decreasing at the same ratio, as branches become narrower, while still adding up to the trunk’s volume. This ‘rule’ had been accepted as part of metabolic scaling theory - which poses that the ability of a living organism to sustain, grow and reproduce depends on the energy available from its environment per unit time.

“While a great tip for artists, which is what Da Vinci intended, Leonardo’s rule of trees does not hold up at the micro level."

Professor Ruben Valbuena, Bangor University and SLU

This had been largely adopted by science when modelling trees and how they function. Now, scientists at Bangor University in Wales, and the Swedish University of Agricultural Sciences (SLU), have discovered that this rule contradicts those that regulate the internal structures of trees. Publishing in the peer-reviewed journal PNAS, the researchers showed that this model is not exactly correct when applied to the internal vascular structures of trees.

For water and nutrients to move efficiently through the tree, from root to leaf-tip, the vascular system has to maintain hydraulic resistance, which is calculated from the viscosity of the fluid and the length, shape, and cross-sectional area of the channel. Researchers Professor Ruben Valbuena and PhD student Stuart Sopp have calculated that for hydraulic resistance to work, there comes a point where the rule of trees can no longer hold true.

In order to efficiently transport liquids from roots to leaf-tips, a tree’s vascular channels need to maintain a certain dimension to maintain hydraulic resistance. Therefore, the plant has to reduce in its volume as it reaches its extremities, causing a higher ratio of capillary to the surrounding plant mass.

As Valbuena explains, “While a great tip for artists, which is what Da Vinci intended, Leonardo’s rule of trees does not hold up at the micro level. We believe our calculations further refine metabolic scaling theory and improve understanding the plant system as a whole.

"Our recalculations may also explain why large trees are more susceptible to drought, and may also be at a greater vulnerability to climate change.”

Sopp said, “One of our aims was to produce a ratio which could be used to estimate tree biomass and carbon in forests. This new ratio will assist in calculating global carbon capture by trees.”