A cube law with a lever arm

The cube law is derived in four lines: pumping cost falls as the fourth power of a tube’s radius, upkeep rises as the square, their sum is least where flow goes as the cube, and conservation at a junction does the rest. The same essay names Da Vinci’s rule — total cross-section conserved, an exponent of two — as “the mechanical answer”, a branch that must hold itself up rather than carry sap. It gives no derivation for that answer, and neither does any essay beside it.

A tree’s radii read against its tip counts can say which exponent a tree has. They cannot say why. The question here is whether a mechanical rule, stated as carefully as Murray’s, gives two — and what it gives when it does not.

Murray’s rule in one line #

Murray’s argument sizes each branch by the flow it carries, and in a tree whose every tip draws the same flow a branch carries flow in proportion to the number of tips it feeds. The optimum makes radius cubed proportional to flow, so r³ is proportional to tips, and since a junction’s tips are its daughters’ tips added, every junction conserves r³.

Notice what does not appear: the length of any branch. Two trees with the same topology and the same tip flows have the same radii under Murray’s rule whether their branches are long or short, straight or bent. That absence is what the rest of this essay turns into a test.

A load on every tip #

The mechanical rule needs its own premises, stated as plainly. One: the same force acts on every tip, as wind does on leaves. Two: every branch is sized so that the bending stress at its base is the same everywhere in the tree. Three: the lever arm of each tip’s force about a branch’s base is the straight-line distance from that base to the tip. Four: the tree is self-similar — symmetric, each generation’s branches λ times the length of the generation before, turned by the same angle at every fork.

Each is an assumption about a tree and none is derived from anything. The third is the one that will be tested by breaking it, and the fourth is the one no real tree meets exactly.

Equal stress means r³ goes as the moment #

A solid circular branch carrying a bending moment M has a peak stress of 4M/(πr³) at its surface. Equal stress everywhere therefore means r³ proportional to M, the same cube as Murray’s rule but of a different quantity.

The moment at a branch’s base is the sum, over every tip it carries, of the force times that tip’s lever arm. So where Murray’s rule makes r³ count tips, the stress rule makes r³ count tips weighted by how far away they are. That weighting is where the lengths come in.

The exponent a junction conserves #

In a self-similar tree the subtree above a generation-g branch is a copy of a k-generation tree, k = D − g, scaled by λᵍ. Distances are unchanged by the turns, so its moment is F·2^(k−1)·λᵍ·Aₖ, with Aₖ the mean lever arm of a k-generation tree in its own units.

A symmetric junction conserves rᵖ when its parent’s rᵖ is twice a daughter’s. With r³ proportional to M, the moment at generation g over the moment at generation g + 1, raised to the power p/3, equals 2, and substituting the moments gives pₖ = 3 / log₂(2·Aₖ / (λ·Aₖ₋₁)) — an exponent that depends on how far the junction is from the tips, through Aₖ.

Deep inside the tree #

For any λ below one the mean arm converges as the subtree deepens, because the lengths form a geometric series, so Aₖ/Aₖ₋₁ tends to one and the exponent tends to p = 3 / (1 + log₂(1/λ)). That is the interior limit, and it is exact there.

Three values make it concrete. A crown whose 2ᵍ branches of generation g fill a plane with area λ²ᵍ each has λ² = 1/2, so log₂(1/λ) = 1/2 and p = 3/1.5 = 2 exactly. One that fills a volume has λ³ = 1/2 and p = 3/(4/3) = 9/4. Halving lengths gives 3/2, and no shortening at all gives three.

Two from a plane #

So Da Vinci’s exponent of two comes out of a stress rule, and it comes out for one geometry: a crown whose branches shrink so as to fill a plane. That is not an arbitrary coincidence of numbers. A crown filling a plane keeps its branches’ total lever-weighted load and their total cross-section in the same proportion from generation to generation, which is what conserving area at every junction requires.

It also means Da Vinci’s rule, read this way, is not a rule about trees in general. It is the answer for trees of one shape, and other shapes want other exponents.

Measured on built trees #

The derivation is checked on trees built branch by branch, with every radius computed from the actual distances to every tip. At the trunk of a fifteen-generation tree the measured exponent is 1.500 at λ = 0.5 against a derived 1.500, 1.727 at 0.6, 1.980 at 0.7 against 1.981, and 2.262 at 0.8 against 2.269.

It falls behind as λ approaches one: 2.558 at 0.9 against 2.604 and 2.803 with no shortening against three. That is not the derivation failing. With no shortening the arms never converge — the lengths form no geometric series — and the limit is approached only logarithmically, so fifteen generations are not deep inside.

The lengths decide the mechanism #

Murray’s flow rule conserves three at every length ratio, because flow does not care how long a branch is. The stress rule conserves 3/(1 + log₂(1/λ)), which moves from one and a half to three as λ moves from a half to one. So two trees with the same topology and the same exponent of three can be told apart by their lengths: one sized for flow can have any λ, one sized for stress must have λ near one.

Put the other way, a measured exponent alone does not name a mechanism, and a measured exponent with the tree’s length ratio beside it does. The quantity that fitting the exponent never recorded is the one that separates the explanations.

The trees drawn beside the cube law #

The branching trees drawn beside the cube law shrink each generation’s branches by 0.74. A tree of those proportions sized for wind would conserve 3/(1 + log₂(1/0.74)) = 2.09, and a seventeen-generation one measures 2.0905. The widths those trees are drawn with, from the cube law, are what that tree’s flow wants; its stress would want widths thinning faster.

The arithmetic is worth doing once by hand. 1/0.74 is 1.3514, whose base-two logarithm is ln 1.3514 / ln 2 = 0.3011/0.6931 = 0.4344. So the denominator is 1.4344, and 3/1.4344 = 2.0915.

That sits beside the finding that the same trees’ fork angles came from a constant. Their lengths were a constant too, and a constant nobody related to either rule — which turns out to be a choice between them.

How precisely the lengths must be known #

A prediction is only as good as its input, and the stress rule’s input is a length ratio. Its exponent moves with λ at dp/dλ = 3 / ((1 + log₂(1/λ))²·λ·ln 2), which is 2.72 at a planar crown, 2.84 at the 0.74 the drawn trees use and 3.07 at a volume-filling crown.

So a length ratio measured to within 0.05 fixes the predicted exponent to within about 0.14. That is a useful comparison: a fitted exponent on fifty good junctions measured to five per cent of radius already carries more uncertainty than that, so a tape measure on branch lengths is not the weak link in testing the rule.

The length ratio an exponent needs #

The formula turns round exactly: a tree conserving an exponent p under the stress rule must have λ = 2^(1 − 3/p). An exponent of 1.8 needs branches 0.630 of their parents’ length; 2 needs 0.707; 2.25 needs 0.794; 2.5 needs 0.871; and 3 needs no shortening at all.

That makes every published exponent a prediction about a tree’s lengths. A tree reported at 2.5 is, if wind sized it, a tree whose branches are about seven eighths of their parents — and if its branches are instead two thirds, wind did not size it, or not alone.

Halving lengths #

With each branch half the length of its parent, the stress rule wants 1.5, and the eight-generation tree drawn above conserves 1.498 at its trunk. That is below Da Vinci’s two: the daughters together have more cross-section than their parent, by a factor of 2^(1/3) = 1.26 at every generation.

A measured exponent of 1.5 on a real tree would read, against the two familiar rules, as a failure of both. Against the stress rule it is the prediction for a crown whose branches halve, and the length ratio is a thing a person can measure with a tape.

The outer generations fall away #

The limit holds deep inside and fails near the tips, where the subtree is too shallow for its mean arm to have converged. For a crown filling a plane the outermost junctions conserve 1.349, those five generations in 1.866, and the trunk of a seventeen-generation tree 2.000. For a crown filling a volume the three are 1.410, 2.018 and 2.247; for halving lengths, 1.181, 1.475 and 1.500.

Every junction in a generation conserves the same exponent, to within two parts in a trillion, because a distance does not depend on which way a subtree leans. The whole departure from the limit is a function of how far from the tips a junction sits.

A tree read at its twigs #

That has a practical edge. The junctions most easily reached on a real tree are its outer ones, and on a stress-sized tree those conserve well below the interior exponent — 1.35 against two for a planar crown. A study that measured only the outer three generations would report something near one and a half and conclude the tree obeys neither rule.

It is the same trap the band that decides the answer found for measurement error, arriving from the mechanics rather than the instrument: where on the tree the junctions are taken decides what the tree appears to obey.

With every fork straight #

If every fork is straight, the lever arm is the path length and the mean arm has a closed form, Aₖ = (1 − λᵏ)/(1 − λ), so the exponent at every junction is known exactly. For a planar crown it is 1.321 two generations above the tips, 1.750 at four, 1.948 at eight and 1.998 at seventeen. For halving lengths it reaches 1.496 by eight; with no shortening, only 2.759 by seventeen.

A planar tree built branch by branch with straight forks matches the closed form at every one of its junctions to three parts in a hundred trillion. That is the check that the built trees measure what the derivation describes, before any fork is turned.

Turning the forks #

Turning the forks shortens the straight-line arms a little, and speeds the convergence a little. At thirteen generations a planar crown’s trunk conserves 1.9911 with straight forks, 1.9931 at a half-angle of 15°, 1.9970 at 30° and 1.9993 at 45°.

So the fork angle, which the cost of a network chooses on its own grounds, barely touches the exponent the stress rule wants. The length ratio carries the exponent; the angle adjusts the approach to it.

How deep is deep #

The trunk closes on its limit from below and never passes it. For a planar crown it reads 1.866 at five generations, 1.980 at nine, 1.992 at eleven and 2.000 at seventeen. For the site’s 0.74 it runs from 1.925 to 2.091. For halving lengths it is already 1.499 at nine.

With no shortening it runs from 2.330 at five generations to only 2.831 at seventeen, against a limit of three. A real tree has perhaps ten or a dozen generations from trunk to leaf, so for most length ratios its trunk would sit within a few hundredths of the limit, and for a tree whose branches barely shorten it would not be near it at all.

Breaking the third assumption #

Take the lever arm as each tip’s height above a branch’s base rather than its distance — a horizontal wind acting on a tree whose branches lean. The trunk exponent moves from 1.997 to 1.755, and the junctions of one generation stop agreeing: the widest range of exponents inside a single generation is 1.425.

That is what a derivation leaning on self-similarity does when the self-similarity is taken away. A height is not a distance, a branch leaning left and one leaning right feel different arms, and a tree sized to one stress under that load has no single exponent at all.

What a real load would need #

A real wind is neither of the two arms. It acts on leaves distributed along branches, not only at tips; it gusts; and a branch deflects under it, changing its own arm. Self-weight is worse: a load spread along every branch, proportional to the wood in it, which depends on the very radii being solved for.

None of those is computed here, and the vertical-arm result says why they matter: a derivation that gives exactly two under one idealisation can give no single answer under a nearby one.

Read by its tip count #

A stress-sized tree can be read the way a count reads a tree, by regressing log radius on log tip count. For a planar crown of eleven generations that reads 1.798 over every branch with at least two tips, 1.922 over branches with at least eight, and 1.967 over those with at least thirty-two, against a limit of two.

The count reads what the tree conserves, and a stress tree conserves less near its tips. Leaving the outer generations out of the regression brings it towards the interior exponent, for every length ratio: from 1.852 to 2.045 for the drawn trees’ 0.74, from 2.219 to 2.592 with no shortening.

This is not new #

The result is not original here, and saying so is part of reporting it. Elastic and stress similarity as explanations of tree form go back to Thomas McMahon’s work in the 1970s, and Christophe Eloy derived Da Vinci’s rule from wind-induced stress in self-similar trees in 2011.

What is measured here beyond the derivation is the exponent generation by generation, how many generations the limit needs, what a turned fork does, what a height-based arm breaks, and what a tip-count regression reads from such a tree — the numbers a person would need to test the idea on a real crown.

What this does not establish #

That any tree is sized for wind, that its tips carry equal loads, that its crown is self-similar, or that self-weight gives the same answer. Nothing here measures a tree. The trees are symmetric, and an asymmetric tree has no single length ratio to put in the formula.

It also does not say Murray’s rule is wrong for trees. A trunk carries sap and bends in wind at once, and a tree sized by both would conserve neither three nor the stress rule’s exponent but something the two costs agree on. What a mechanism would have to show is a derivation with its parameters measured, and λ is a parameter nobody has yet put beside a fitted exponent.

What would withdraw it #

A deep stress-sized tree whose trunk does not converge on 3/(1 + log₂(1/λ)). A straight-forked tree that disagrees with the closed form at any junction. Flow-sized radii on the same trees that conserve anything but the cube. Each is checked every time the measurement runs.

A load that depends on the answer #

The next test takes away the idealisation that made the derivation clean: a load carried by the branches themselves, proportional to the wood in each. Under self-weight the moment at a branch depends on the radii of every branch above it, so the sizing is a fixed point rather than a formula, and the test is whether it still conserves a single exponent set by the length ratio — and, if it does, whether a planar crown still gives two.