A crown sized for how far it bends

A crown that carries its own wood sized three trees and held one thing fixed in all of them: the bending stress at the base of every branch. Under loads on the tips that gave Da Vinci’s exponent of two for a crown filling a plane; under the crown’s own weight it gave one; under wind on the wood it gave two again, as a limit approached slowly. The essay ended by naming the criterion it had not tried.

Stress is what breaks a branch. Stiffness is what a branch does before it breaks, and a tree that bends its leaves into its neighbour’s light has lost something without breaking anything. McMahon’s elastic similarity holds the deflection fixed instead — a branch is sized so that its own weight bends it by a fixed share of its length — and for a single beam that gives radius as length to the three halves rather than to the square.

What is held fixed, and what that does to the arithmetic #

A branch is treated as a cantilever fixed at its base, with every load in its subtree acting perpendicular to the arm from that base. That is the same idealisation a cube law with a lever arm used, carried unchanged, so the only thing separating that account from this one is what is held equal.

The end of a cantilever of length $L$ under a load $P$ at arm $a$ deflects by $P a^{2}(3L - a)/6EI$ for a load on the branch, and by $P L^{2}(3a - L)/6EI$ for one beyond it, where $EI$ is the bending stiffness and the second moment of area goes as $r^{4}$. Holding the deflection as a share of the length equal across branches therefore sizes

$$r^{4} \propto \frac{1}{L}\sum P \cdot K(a, L)$$

where equal stress sized $r^{3}$ against the plain sum of $P a$. Two things change together: the radius enters as its fourth power rather than its third, and the arm enters through a kernel that is quadratic in it rather than linear.

Why a deflection is divided by a length #

The criterion could have been written as equal deflection rather than equal deflection ratio, and that would be a different and worse rule. A twig that hangs four millimetres and a limb that hangs four millimetres are not in the same condition: the twig has moved a third of its own length and the limb a hundredth. Holding the absolute sag equal would make every twig enormously overbuilt, which is not what trees look like.

Dividing by the branch’s own length is what makes the criterion self-similar, and self-similarity is what gives it a single exponent to conserve. The same choice sits under the stress rule, where the quantity held fixed is a stress rather than a force for exactly the same reason — and under Murray’s flow rule, where it is the work per unit volume of fluid rather than the work.

Two changes that do not cancel #

That is worth pausing on, because a change of exponent on both sides of an equation often does nothing at all. It does not cancel here. Raising the radius power makes the tree thin more slowly from tip to trunk for a given load pattern; making the arm count more makes the loads at the ends of long branches matter more. Which of the two wins depends on how fast the branches shorten, and that is exactly why the answer has a length ratio in it.

The exponent a deep crown conserves #

Deep inside a self-similar crown, a branch $k$ generations above the tips carries $2^{k}$ tip loads at arms proportional to its own length $\lambda^{-k}$. So $r^{4}$ goes as $2^{k}\lambda^{-2k}$, the radius ratio from one generation to the next is $(2\lambda^{-2})^{1/4}$, and a junction conserves $r^{p}$ with

$$p = \frac{4}{1 + 2\log_{2}(1/\lambda)}$$

against the stress rule’s 3/(1 + log₂(1/λ)). Writing ℓ for log₂(1/λ) — how many halvings of length each generation is worth — the two are 4/(1 + 2ℓ) and 3/(1 + ℓ).

The crowns, drawn #

At the planar length ratio the two drawings are the same tree, which is the result this essay is mostly about. At a crown whose branches halve they are visibly different, and the equal-deflection crown is the one with the thinner trunk.

Measured rather than derived #

The closed form is a deep-crown limit, so the crowns were built and their junctions read one generation at a time. At the trunk of a fifteen-generation crown the exponent is 1.3333 where the limit is 1.3333, 1.4679 where it is 1.4679, 1.6168 where it is 1.6169, and 1.9990 where it is 2 — every one within two thousandths. Every junction within a generation conserves the same exponent to within 4 × 10⁻¹⁴, which is the arithmetic’s own noise: a distance does not care which way a subtree leans.

And the outer junctions are not the crown #

The junctions near the tips conserve a smaller exponent, and the gap is large: at the planar ratio the outermost junctions read 1.3411 against a limit of 2, reaching 1.9722 only eight generations in. The reason is that the arm from a junction to its tips has not yet settled into proportionality with the branch’s own length when there are two tips rather than two thousand.

That happens under the stress rule too, and it happens over a longer run here, because the arm enters this criterion squared. It is the single most important practical fact in the account: a measured tree has few generations, and the exponent it reads is not the limit.

Seven generations is not deep #

A crown of seven generations reads 1.9553 against a planar limit of 2, and 2.2925 against a volume-filling limit of 2.4 — under by two and by four per cent. Every ladder closes from below and none of them arrives. So a trunk exponent measured on a real crown is a lower bound on its own rule’s limit, and the shortfall is larger for crowns whose branches shorten slowly.

The one ratio where the two criteria agree #

Setting the two closed forms equal gives 3(1 + 2ℓ) = 4(1 + ℓ), which is 3 + 6ℓ = 4 + 4ℓ, whose only root is ℓ = 1/2. That is λ = 2^(−1/2), the crown that fills a plane, and there both exponents are 4/2 = 3/1.5 = exactly two.

So the two criteria size the same crown at one length ratio in the whole family, and it is the ratio Da Vinci’s rule names. That is not a coincidence arranged by the algebra: the same exponent falls out of holding stress fixed and holding deflection fixed, at the one crown whose branch lengths halve in area rather than in length.

Checked by sweeping rather than by solving #

A sweep of two hundred length ratios from 0.35 to 0.99 finds exactly one sign change in the difference, within a hundredth of the derived crossing, with the difference negative at every ratio below it and positive at every ratio above. Below the planar ratio the stiffness rule sizes the thinner trunk and above it the thicker: at λ = 0.5 the two differ by 0.1667 and at λ = 0.9 by 0.4633.

What the disagreement is for #

A crossing is only interesting if the curves are far apart somewhere else, and they are. That is what makes the two criteria distinguishable on a measured tree, and it is the subject of the next essay: three rules, one measured exponent, and what a length ratio has to be known to before the exponent decides between them.

What a fork’s angle has to do with it #

Nothing in the sizing reads the fork’s half-angle, and the crowns here are drawn at thirty degrees because that is what the trees drawn at no angle established the default should be. The angle does enter the arms — a wider fork puts a subtree’s tips further from the junction it hangs from — but it enters every branch’s arm in nearly the same proportion, so what it moves is how fast the exponent converges rather than what it converges to.

Measured, a thirteen-generation planar crown reads 1.992601 at a straight fork, 1.994227 at fifteen degrees, 1.997454 at thirty and 1.999378 at forty-five: all below two, all within a hundredth of it, and a wider fork closer. The whole spread across a straight tree and a right-angled one is 0.0068, against the 0.6563 the same crown’s junctions span between its tips and its trunk. So the angle is a hundredth of the effect the depth is, which is why the length ratio is the parameter this family is written in.

It is also the reason this account and the fork-angle work are about different things. A fork’s angle is settled by what happens at the junction; a crown’s exponent is settled by what happens along the branches, and the two minimisations do not talk to each other.

The single beam, where the law has a name #

McMahon’s result is about one beam, and it drops out of this arithmetic rather than being assumed by it. A cantilever carrying its own weight has a load per unit length going as $r^{2}$, so its end deflection is proportional to $r^{2}L^{4}/r^{4}$ and its deflection as a share of its length to $L^{3}/r^{2}$. Holding that fixed gives $r \propto L^{3/2}$.

The same beam held to a stress instead has a base moment going as $r^{2}L^{2}$, divided by $r^{3}$, so $\sigma \propto L^{2}/r$ and $r \propto L^{2}$. A tip load held to a deflection gives $r \propto L^{1/2}$ and held to a stress $r \propto L^{1/3}$.

Fitted, because the self-weight beam is a fixed point #

The four exponents were not written down; they were fitted from beams built at seven lengths spanning a factor of eleven, and the fitted slopes come out at 0.333333, 0.500000, 2.000000 and 1.500000, each straight to within 2 × 10⁻¹⁴ in log radius. That matters for the two self-weight cases, because a beam carrying its own weight is a fixed point rather than a formula — the load depends on the radius being solved for — and a fixed point solved wrongly is the kind of error that produces a plausible exponent.

The wood, where the crown becomes a fixed point too #

Moving the load off the tips and onto the branches makes the whole crown a fixed point, exactly as it did under the stress rule. The deep-crown limits follow from the same self-similarity: a branch dominated by its own weight has its deflection ratio going as $r^{2}L^{3}/r^{4}$, so its radius goes as $L^{3/2}$ — the single beam’s law, recovered inside a crown — and the junctions conserve $(2/3)/\ell$. Under wind on the wood the radius goes as $L$ and they conserve $1/\ell$.

Four thirds where the stress rule read one #

At the planar ratio the weight limit is (2/3)/(1/2) = 4/3. A thirteen-generation crown reads 1.3157 and a fifteen-generation one 1.3231, closing from below. Under the same load the stress rule read one — radius rather than area, the stress-similarity law that radius goes as length squared.

So the criterion changes what a crown’s own wood does to it, and by a third rather than by a rounding. A crown whose trunk is dominated by the weight of its own branches conserves radius if it is sized for strength and something distinctly above it if it is sized for stiffness, and those are different trees. The handover between the two, on a crown carrying leaves at its tips and wood along its branches, is set by the share of the trunk’s load the wood carries — which is the reading a crown that carries its own wood made under the stress rule, and which runs the same way here with a different pair of endpoints.

Where the wood regime stops #

Both limits have a length ratio above which they stop holding, and the reason is the same as under the stress rule. The load a branch’s subtree contributes relative to its own is a geometric series, and the series converges only while the branches shorten fast enough: $2\lambda^{4} < 1$ for weight, so $\lambda < 2^{-1/4}$; $2\lambda^{2} < 1$ for wind, so $\lambda < 2^{-1/2}$, which is the planar crown exactly. Above those the load is dominated by the outermost twigs, the crown behaves as if the load were on the tips, and the exponent is the tip-load one.

What a measurement of this would look like #

A trunk exponent is read off a set of junctions, and fitting the exponent is the account of how that goes wrong: the estimator is biased, the bias depends on which junctions are in the sample, and a junction whose daughters are nearly equal carries almost no information about the exponent at all. Which junctions say anything is the sharper version — the informative junctions are the lopsided ones, and they are the minority.

None of that changes here, and one thing about it gets worse. Both criteria in this essay predict an exponent that is approached from below with depth, so a sample drawn from the outer parts of a crown — which is where the accessible junctions are — reads lower than the limit under either rule. Two biases pointing the same way are harder to argue with than one, and the correction is the depth ladder above rather than anything about the estimator.

What the criterion does not decide #

Which of the two a tree is actually sized by is not settled by any of this, and nothing here measures a tree. Both criteria are optimisation arguments about a symmetric crown with one length ratio, and a real crown has neither. Both ignore buckling, which is a third criterion with its own exponent; both ignore that a branch’s material is not isotropic and its cross-section is not a circle; and both treat a static load where the thing that actually breaks branches is usually dynamic. And both are optimisations, which means they inherit the difficulty an optimum too flat to reach names: a minimum broad enough that a wide range of trees sits within a per cent of it is a minimum a plant has no pressure to find exactly.

What the pair of results does establish is narrower and harder to argue with: the exponent a crown conserves is a property of what is held fixed as well as of how its branches shorten, and the two criteria coincide at one length ratio out of the whole family.

Where this is not new, and what is #

Elastic and stress similarity in trees are McMahon’s, from the 1970s, and the three-halves law for a single beam is his result rather than one found here. Deriving Da Vinci’s exponent from wind-induced stress in a self-similar crown is Eloy’s, from 2011. Both are standing results and neither is being claimed.

What this adds is the pair put on one axis. The stiffness criterion has been applied to single beams and to whole trunks; what had no number attached to it was the exponent a junction conserves under it, generation by generation, as a function of the length ratio — and therefore whether the two criteria ever agree. They do, at one ratio, and it is the ratio the oldest rule in the subject names. That is the sort of statement that is either true or false rather than plausible, and it took building the crowns to find.

What is claimed, in one line #

A crown whose every branch bends by the same share of its own length conserves 4/(1 + 2·log₂(1/λ)) at its interior junctions, reduces to McMahon’s three-halves law on a single beam under its own weight, and agrees with the equal-stress crown at exactly one length ratio — the crown that fills a plane, where both give two.

What would withdraw it #

A trunk exponent that is not 4/(1 + 2ℓ) to two thousandths at fifteen generations. Two junctions in one generation conserving different exponents. A fitted single-beam slope that is not 3/2, 2, 1/2 or 1/3. A second sign change in the difference between the two exponent curves. A planar crown under its own weight that does not climb towards four thirds. Each is checked every time the measurement runs.

Still open: a crown that would rather not buckle #

Stress and stiffness are two of the three ways a branch fails, and the third has no exponent here. A slender column under its own weight buckles at a critical height going as $r^{2/3}$, so a branch sized to stay a fixed fraction below its buckling height has radius as length to the three halves — the same power as elastic similarity, arrived at from a different failure. Whether that is a coincidence of the single beam or survives into a crown is the next measurement: the buckling criterion applied junction by junction, and whether it conserves any single exponent at all, given that a branch’s critical load depends on the compression its parent carries and therefore on the whole path to the ground.