Misleading title, the article body goes on to say that a different rule other people came up with, that was an assumption based on Leonardo da Vinci's rule, is the one that's been disproven. Says nothing about his actual rule.

Isn’t it also incredibly easy to prove Leonardo’s tree rule just by cutting down a few?

You can't prove something by example, you can only show that the pattern holds for the examples that you've checked.

Biology isn't mathematics.

Yes, in math you can't prove things by example.

But in biology that's the only way you form knowledge.

Biology doesn't have axioms. In biology, proof is by induction. However philosophically unsatisfying that may be.

Alternatively, you don’t “prove” things in biology, you just provide evidence for them.

Proof implies a level of certainty that doesn’t exist in biology.

In fact you don't prove anything in science. You merely fail to disprove something.

in the “natural sciences”

Do we have innatural ones?

Is computer science a science?

Of course not. It's a branch of mathematics. Always has been.

„A branch“ - so we are back to trees then?

Mathematics and computer science are both in the category of formal science.

Yes, it’s a formal science.

The other category of science is called formal science.

Is proving something internal to a system of ad-hoc rules that cannot (to the same standard of proof) be proven to have any correspondance with the natural world really proving something? Sincere question.

I think so, yes. You start with axioms, and the proof follows from those.

You can argue whether that’s interesting or not, but IMO it’s as much a proof as anything can be, and IMO it’s very interesting.

On the other hand, "trees" are polyphyletic, among other aspects of their broad diversity. So even if you can generalize across a species, generalizing across all trees from a handful of examples is likely to lead you astray.

On the gripping hand, trees are polyphyletic because the constraints of physics and biology push in a certain direction regardless of starting point; you can imagine there is some underlying mathematical rule to this abstract "tree" even if it isn't realized in all examples.

You can imagine, sure, but we were talking about proof.

Now do the same with dynamic types and unit tests.

Yea I misspoke. (typed)

I think that's correct for some assertions.

When a statement is "this is always true", then a counterexample is all it takes to disprove it.

On the other hand, if the statement is "this is not always true" then an example of it happening is as you said - the example does not counter anything.

> ... Says nothing about his actual rule.

Indeed! Here it is in brief from another article on the subject

"...Leonardo’s rule [for how trees branch] says that the thickness of a limb before it branches into smaller ones is the same as the combined thickness of the limbs sprouting from it."

https://www.sciencenews.org/article/leonardo-da-vinci-rule-t...

That seems to again misrepresent the rule. He’s an artist drawing 2D objects not making 3D sculptures.

The connection between the surface area of branches and overall tree structure shows that it’s the living, outer layers that guide tree structure, the researchers say. “The life of a tree flows according to the laws of conservation of area in two-dimensional space,” the authors write in their study, “as if the tree were a two-dimensional object.”

But now we know that most trees are living skin on a dead skeleton. If we assume a constant thickness of the vascular cambium, branches must conserve surface area, and not volume! Now, off to read the paper.

Edit: aha, my proof is correct! Only, my axiom was wrong: trees make broader capillaries for thinner branches. Well played, trees.

...or, to quote the article:

> 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

The way I read it: because branches at the extremities are thinner, the ratio of "living skin" to "dead skeleton" is higher. I'm just wondering why this isn't immediately obvious and why they are presenting this as a great discovery?! Also, the phrase "has to reduce in its volume as it reaches its extremities" sounds a bit backwards: sure, young branches are thinner, and as the tree grows and branches get older, they grow thicker to support the young branches that grow off them.

> If we assume a constant thickness of the vascular cambium, branches must conserve surface area.

Not necessarily. Transport speed could be different depending on the diameter of the branches

Also, I would think a model that has a constant flow at each distance from the roots or the leaves is incorrect.

A more realistic model has water being transported up and mostly evaporating in the leaves and barely contributing to tree mass, carbon (in some form) being transported down from the leaves and gradually being accumulated, and small amounts of minerals being transported up from the roots and gradually being accumulated.

That model has a constant water flow throughout the tree, but a gradient in downward flow. I wouldn’t know which is dominant.

_if_ most of the transport volume is carbon being transported down to branches and roots and being accumulated, more volume would go out of the leaves than arrives at ground level.

I wouldn’t know whether that’s true, though. Most of a tree’s mass comes out of CO2 in the air, but because water evaporates out of the tree, there still may be more stuff going up than going down.

So a rule generalised from a 500 year old art guideline does not hold up on the micro scale in 21st century science? I'm astonished it even got that far.

Right. Like the Newtonian physics of trees.

I have been looking at trees my entire life, and never thought to even think about this “plumbing” ratio. What a lovely example of how Leonardo combined mathematics and observations.

The body says large trees are more susceptible to drought, but the first companion article in the list below the article says large trees are less susceptible to drought. There needs to be a reconciliation.

A fascinating book in this context is "The Artistic Anatomy of Trees" (1920) by Rex Vicat Cole. It made me look at trees with a lot of new insights.

The illustrations in the book are pretty crappy, but the text more than makes up for it. Highly recommended to people who like to draw or paint from their imagination.

https://openlibrary.org/books/OL23278992M/The_artistic_anato...

Sensationalist title? Seems to me that it was just a heuristic and not really a mathematical axiom.