Leonardo da Vinci's tree rule may be explained by wind

Leonardo da Vinci's tree rule may be explained by wind
(Left) A model of tree branching. (Middle) A tree skeleton with all branches having the same thickness. (Right) The same tree with branch diameters calculated from a model accounting for wind-induced stress, which closely follows Leonardo’s rule. Image credit: Christophe Eloy. ©2011 American Physical Society

(PhysOrg.com) -- More than 500 years ago, Leonardo da Vinci observed a particular relationship between the size of a tree’s trunk and the size of its branches. Specifically, the combined cross-sectional areas of a tree’s daughter branches are equal to the cross-sectional area of the mother branch. However, da Vinci didn’t know why tree branching followed this rule, and few explanations have been proposed since then. But now in a new study, physicist Christophe Eloy from Aix-Marseille University in Aix-en-Provence, France, has shown that this tree structure may be optimal for enabling trees to resist wind-induced stresses.

In his study, which is published in a recent issue of , Eloy explains that Leonardo’s rule is so natural to the eye that it is often used in computer-generated trees. Although researchers have previously proposed explanations for the rule based on hydraulics or structure, none of these explanations have been fully convincing. For instance, the hydraulic explanation called the “pipe model” proposes that the branching proportions have to do with the way that vascular vessels connect the tree’s roots to its leaves to provide water and nutrients. But since vascular vessels account for as little as 5% of the branch cross section (for large trunks in some tree species), it seems unlikely that they would govern the tree’s entire architecture.

“The usual textbook explanation for Leonardo's rule (and, more generally, for the relation between branch diameters) involves hydraulic considerations,” Eloy said. “My study shows that an alternative explanation can be given by considering external loads, such as wind-induced forces.”

Eloy has proposed that Leonardo’s rule is a consequence of trees adapting their growth to optimally resist wind-induced stresses. It’s well-known that plants can alter their growth patterns in response to mechanical sensation, such as wind. The phenomenon, called “thigmomorphogenesis,” means that wind can influence the trunk and branch diameters of a tree as its growing. The underlying cellular mechanisms of this phenomenon are largely unknown.

Building on this line of thinking, Eloy used two models to predict the probability of a fracture at a certain point in a tree due to strong winds. He found that, when the probability of fracture is the same everywhere on the tree, so that each part bears the stress equally, Leonardo’s rule is recovered. He also showed that the diameters of each branch on a tree can be calculated by knowing the parameters of a simple tree skeleton.

Although some of the most common tree species, such as maples and oaks, seem to follow Leonardo’s rule, there are many species that don’t follow the rule, and many more that scientists have yet to analyze.

“Actually, Leonardo’s rule has not been assessed for that many species,” Eloy said. “So far, it seems to be hold for about 10 species. The problem is that it takes a lot of time to measure a single tree, which has thousands of branches, and the data are usually very scattered. Besides, some species clearly do not satisfy Leonardo's rule, such as baobabs, koas, and most bushes.”

The finding that trees seem to follow Leonardo’s when adapting their growth to tolerate wind-induced stresses could have applications both in nature and technology.

“It has obvious applications to the forestry industry to calculate the yields of tree stands and to evaluate the risks of breakage during storms,” Eloy said. “It could also be applied to manmade branching structures such as antennas.”

He added that there is still much more to understand about tree design, including the self-similarity shared by large trunks and smaller branches.

“I am still working on this subject, in particular to try to relate growth to external loads,” he said. “In other words, I would like to understand the dynamical growth mechanisms that lead to the intricate fractal structures of .”

Explore further

Uncovering Da Vinci's rule of the trees

More information: Christophe Eloy. “Leonardo’s Rule, Self-Similarity, and Wind-Induced Stresses in Trees.” Physical Review Letters 107, 258101 (2011). DOI: 10.1103/PhysRevLett.107.258101
Journal information: Physical Review Letters

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Jan 04, 2012
there is still much more to understand about tree design, including the self-similarity shared by large trunks and smaller branches.

Well, it's an architecture which requires almost no "blue print". You simply repeat the same basic shapes and relationships over and over again, which is from a certain point of view more efficient than the architecture of animals, which in most cases is less fractal in design.

I think this is also related to the breaking mass of a branch when considering gravity. The thicker a branch, the heavier it is, but also the stronger it is, but increasing length of a branch only adds weight, not strength.

Branching increases surface area and mass, which makes a tree weaker to wind and gravity, BUT increases the amount of moisture and sunlight available both to the bark and the leaves. It's a trade-off between energy and moisture absorbtion vs structural integrity of individual parent branch structure.

Jan 04, 2012
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Jan 04, 2012
Brings back memories of writing code to generate Lindenmayer system type fractals from simple axioms, for plant models; very elegant and allowed much control.

Jan 04, 2012
Doesn't seem to apply to Bradford Pears, Southern Oaks and some others I have seen. They also seem to do well in winds! Way to many assumptions, without scientific method. An overly broad thesis. A true study of a large and wide variety of species would be of value however. He does state that it does not seem to apply to all species, but that greatly undermines the initial thesis.

Jan 04, 2012
Recent wind storms in Southern California have destroyed a lot of trees in the Los Angeles Arboretum. They could supply a lot of information. They may have said that a lot of Bradford Pears fell. Also a lot of semi tropical trees. The winds were severe.

Jan 04, 2012
IMHO, if the "rule" does not apply to all true trees, then the use of the word is a misnomer. I'd prefer "Leonardo's pattern" if he was the first one to observe and describe it. Clearly it is not a "rule".

Jan 04, 2012
A lot of trees that are grafted to rootstock these days are grafted to different species, ie, pear is grafted on quince roots. Pears typically come from windier environments than quinces, so I could see drawbacks.

Jan 04, 2012
That cannot be said of southern oaks, Sequoias, Redwoods, etc.

Jan 05, 2012
Actually, the branch to trunk ratios have much more to do with the hydraulics of moving sap through the tree than wind loads.

Jan 05, 2012
Actually, the branch to trunk ratios have much more to do with the hydraulics of moving sap through the tree than wind loads.

"But since vascular vessels account for as little as 5% of the branch cross section (for large trunks in some tree species), it seems unlikely that they would govern the trees entire architecture."

The tree is not a weather man nor a plumber. It's probably just that the thicker parts of the tree have passed more nutrients during it's lifetime than the thinner parts,.. so obviously the trunck would be the thickest. This is also why lunch ladies are fat.

Jan 05, 2012
This is the second time this article has been posted. Why the re-post ?
Just because vascular vessels account for only 5% of trunk diameter doesn't mean the other 95% is independent of vascular flow.
Also, suggesting wind stress adaptation as an alternate reason without considering the relationship between trunk length and branching ratio, as well as leaf profile is not very scientific.

Jan 05, 2012
Nice one; @Returners constant (Re) is directly testable, and most likely linear. (Re)*weight/length = f(length)

the expected linearity constant also STRONGLY shows a blueprint of sorts, by natures design standard (ADAPTATION --WHAKAPAPA). (for tall trees its more obvious). but even non linearity implies design.

the idea also implies and ability of (Re) coherence in an intrinsic part of the plant (perhaps in the entire plant). (I recall a centrifugal (or was it centripetal?) experiment that proves this is the case, for small shrubs)(how ever, roots coherence unknown? we weren't allowed to destroy the experiment)

This might be basic phys 101, but the question "what is the nature of Re coherence?" has far reaching implications that IMO ask one to correctly define quantum gravity.

Jan 07, 2012
Ha ha ha! Bradford pears do NOT do well in wind ( due to inherent weakness of the branch unions)

Jan 08, 2012
Stiff winds typically do not occur in the thoracic cavity nor in the fruit fly egg, yet lungs and trachea exhibit just such a tissue structure. Trees, flies and people are all made of cells. Tissue interactions regulated by genes explain these phenomena.

Jan 08, 2012
Seems to me that leaves play a bigger part in wind stress than thickness and weight. Did Leonardo overlook the wind resistance of leaves, or have you misinterpreted his original thoughts. Having been through four hurricanes, I can vouch for the importance of leaves in the wind. -bierdo

Jan 09, 2012

The wind loading proportional to the accustomed loading of the tree, as derived by the anchorage strength of the root system, is proportional to the divisions of restraint to stress through the system.


Jan 16, 2012
bierdo, I remember a long time ago reading an article, I think in Scientific American mag, where the author explained how the shape, size, and other features of tree leaves of different species could be related to the the need to avoid excessive wind drag. He showed through diagrams how all the various kinds of leaves tended to curl themselves into a tube shape or fold along their length into a kind of V shape.

The fractal nature of tree shapes also, is fairly clear. I think this just indicates that growth of plants involves some processes which get repeated, ie switched on and off, as the newly grown cells adapt to their particular environment [including location within the plant]. In some way the outcome of the process each time is fed back through the system; ie their growth entails a form of self-referencing algorithm.

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