A crown that carries its own wood

A cube law with a lever arm derived Da Vinci’s exponent from a mechanical rule: size every branch so that the same load on every tip bends it to the same stress at its base, and a junction conserves $r^p$ with $p = 3/(1 + \log_2(1/\lambda))$, where $\lambda$ is how much shorter each branch is than its parent. A crown that fills a plane, $\lambda = 2^{-1/2}$, gets exactly two.

The load in that derivation sits on the tips, as wind on leaves. It ended by naming the idealisation to take away next: a load carried by the branches themselves, proportional to the wood in each. Under such a load a branch’s moment depends on the radii of every branch above it, so the sizing is no longer a formula. The question is whether it still conserves a single exponent set by the length ratio, and whether a planar crown still gives two.

A load that depends on the answer #

Stress at the base of a solid branch goes as its bending moment over the cube of its radius, so equal stress everywhere means $r^3$ proportional to the moment. With the load on the tips the moment is a sum of tip forces times their arms, and the arms are fixed by the geometry, so every radius follows at once.

With the load on the wood that stops being true. A branch’s weight goes as $r^2$ times its length, and the wind on it as $r$ times its length, so the moment at the base of any branch is a sum over every branch above it of a quantity containing that branch’s radius. The radii appear on both sides. The only way to size such a tree is to guess the radii, compute every moment, cube-root them into new radii, and repeat until nothing moves.

Three loads and one arm #

Three loads are sized for, separately and together. A load on the tips, the lever-arm derivation’s. The wind on the wood, a force along every branch proportional to $r$ times length. The weight of the wood, a force along every branch proportional to $r^2$ times length.

Every load is taken with the same arm the lever-arm derivation used: the distance from the base of the branch being sized to where the load acts, as if each load pushed perpendicular to that arm. That was the idealisation for wind on leaves, and it is carried unchanged so that the only thing changing is what carries the load. For gravity it is the worst case rather than the real moment — a symmetric crown’s weight exerts no net moment at all on a vertical trunk, because its two sides cancel — and the account below is a statement about that worst case.

The fixed point exists and is found #

The iteration settles quickly. On an eleven-generation planar crown it stops moving to a part in a hundred billion after 66 rounds under weight, 25 under wind on the wood and 15 under tips and weight together. Started instead from every radius equal, it reaches the same tree up to its overall scale, to sixteen decimal places under weight alone.

With the wood carrying no load the fixed point is the lever-arm tree exactly, every radius the same to $2 \times 10^{-16}$. So the fixed-point calculation reproduces the result it extends before it says anything new, and what it says about loads on the wood is a change in the load and in nothing else.

What a branch’s subtree carries #

The deep-tree limits have closed forms, and the argument is a sum. Suppose the radius ratio from one generation to the next settles at ρ. The load on a branch’s whole subtree is its own branch’s load times $\sum_m (2\rho^a\lambda)^m$ over the generations above it, with $a = 2$ for weight and $a = 1$ for wind: each generation up doubles the branches, multiplies each radius-power by $\rho^a$ and each length by $\lambda$.

If that series converges the subtree’s load is dominated by the branch itself, the arm scales with the branch’s own length $\lambda^g$, and $r^3 \propto r^a \lambda^{2g}$. For weight that gives $\rho = \lambda^2$ and an exponent $p = \log 2 / \log(1/\rho)$ of $1/(2\log_2(1/\lambda))$. For wind it gives $\rho = \lambda$ and $p =$ $1/\log_2(1/\lambda)$. The weight’s series converges while $2\lambda^5 < 1$, the wind’s while $2\lambda^2 < 1$.

Under its own weight, a crown conserves radius #

At a planar crown the weight’s closed form is 1/(2 × 0.5) = one. Every junction conserves $r^1$: a parent’s radius is the sum of its daughters’ radii, not its cross-section the sum of theirs.

The arithmetic is worth doing once. With $\rho = \lambda^2 = 1/2$, each daughter has half its parent’s radius and so a quarter of its cross-section, and two daughters carry half the parent’s area between them. A crown sized for its own weight loses half its cross-section at every fork. Da Vinci’s rule keeps all of it and Murray’s rule, at $2^{-1/3} = 0.794$ a generation, keeps 1.26 times the parent’s area between the two daughters.

Radius proportional to length squared is not a new law. It is the stress-similarity rule for a beam carrying its own weight, from the analyses of tree form Thomas McMahon published in the 1970s; what is measured here is what it means for the junction exponent, generation by generation.

Straight trees hold the closed form #

A tree with every fork straight is the cleanest check, because its arms are exactly its path lengths. On a fifteen-generation straight tree at λ = 0.5 under its own weight the inner eight generations conserve 0.500 to within two parts in a hundred thousand of the closed form’s $1/(2\log_2 2) = 0.5$. At λ = 0.6 they conserve 0.678, against 1/(2 × 0.7370) = 0.6785, to within two thousandths.

The planar straight tree converges more slowly: 0.994 at its trunk, falling to 0.968 six generations out and 0.650 at its outermost junctions. Its series ratio $2\lambda^5 = 0.354$ is larger than 0.5’s 0.0625 or 0.6’s 0.156, and a series closer to its critical value takes more generations to settle.

Where the weight hands over to the twigs #

The weight’s series stops converging at $2\lambda^5 = 1$, a length ratio of $2^{-1/5} = 0.8706$. Beyond it each generation’s wood outweighs the one below, the load is dominated by the outermost twigs, and the twigs’ weight acts like a load on the tips — so the exponent becomes the lever-arm derivation’s.

The two formulas meet exactly there. The weight’s gives 1/(2 × 0.2) = 2.5 and the tip load’s gives 3/(1 + 0.2) = 2.5. So under its own weight a crown conserves an exponent that rises from one at a planar crown through 1.151 at the drawn trees’ 0.74 and 1.5 for a crown filling a volume to 2.5 at 0.8706, and then follows the tip load’s curve to three.

Wind on the wood #

The wind’s series is critical at $2\lambda^2 = 1$, which is exactly a planar crown. Below it the exponent is $1/\log_2(1/\lambda)$: one for lengths halving, where the measured trunk of a fifteen-generation tree reads 0.999. Above it the exponent is the tip load’s. At a planar crown both formulas give exactly two.

So Da Vinci’s two survives moving the wind from the leaves onto the wood, at a planar crown and only there as a boundary between two regimes. What it does not survive is the depth of the tree: at the critical point the series grows by one generation’s worth per generation rather than converging, and the approach to the limit is logarithmic.

Two limits, two speeds #

Under its own weight a planar crown’s trunk conserves 0.969 at nine generations, 0.983 at eleven, 0.991 at thirteen and 0.995 at fifteen: within a hundredth of one by thirteen generations, closing geometrically.

Under wind on its wood the trunk conserves 1.629 at nine generations, 1.695 at eleven, 1.743 at thirteen and 1.778 at fifteen. Each two generations add less than the two before — 0.067, 0.048, 0.035 — and fifteen generations leave it 0.22 short of two. A planar crown a dozen generations deep, sized by wind on its wood alone, would read nearer 1.7 than 2 at its trunk, and the difference would be depth rather than error.

Generation by generation #

Read junction by junction on a thirteen-generation planar crown, the weight-sized tree conserves 0.991 at its trunk, 0.969 four generations out, 0.921 at seven, then 0.844, 0.774 and 0.653 at its outermost junctions. The wind-sized tree conserves 1.743, 1.629 four out, 1.461 at seven, then 1.255, 1.089 and 0.835.

Both fall away at the tips for the same reason the lever-arm tree did: an outer branch’s arm has not converged to its deep-tree value. What differs is how far in the falling reaches. For weight the interior holds its exponent to within a tenth for eight generations; for wind the exponent never stops changing across the tree, which is the depth effect read sideways.

Leaves and wood together #

A real crown carries both: leaves at the tips, which catch the wind, and wood along every branch, which weighs. Put a unit load on every tip of a thirteen-generation planar crown and add its weight at increasing strength, and the result is not an average of two and one but a crown that is each in a different place.

With the wood carrying 19.5 per cent of the trunk’s moment, the trunk conserves 1.882 and the exponent peaks at 1.95 halfway out. With the wood carrying 79.6 per cent, the trunk conserves 1.459, the exponent climbs to 1.83 seven generations out where the wood carries only 13 per cent of the load, and then falls away at the tips. With the wood carrying 99.7 per cent, the trunk conserves 1.089 and the exponent climbs to 1.51 nine generations out.

The handover follows the wood’s share #

The wood’s share of a branch’s moment falls outward along the crown, because a twig carries little wood above it and its leaves are close. So the weight’s exponent of one governs where the wood carries the load and the leaves’ two governs where the leaves do, and the passage between them sits where the two shares cross — a place fixed by the ratio of leaf load to wood load, not by any property of the junctions.

Across nine strengths of the weight the trunk’s exponent falls steadily with the wood’s share: 1.959 at 6.5 per cent, 1.723 at 45, 1.459 at 80, 1.237 at 96, 1.089 at 99.7 and 0.999 once the wood carries all of it. The junctions three generations from the tips hardly move while the wood carries up to four fifths of the trunk’s moment — 1.781, 1.771, 1.747 — then read 1.683 at 96 per cent and 1.514 at 99.7, following the trunk down only once the wood carries nearly everything.

What that does to a measured exponent #

A single exponent fitted to such a crown is a weighted average over junctions that conserve different things. A count of the tips each branch carries weights every segment, the many twigs among them; a junction fit weights the informative junctions, wherever the forks are even. Nothing makes those two averages agree on a crown whose exponent changes from trunk to twig, and nothing here measures by how much they differ.

That is a caution for every comparison between a fitted exponent and a rule. A reading near 1.6 is what a tree at Da Vinci’s two gives a junction fit at twelve per cent of error, what a count gives a tree at two that has lost seven tenths of its tips, and what the trunk of a planar crown gives when its wood carries somewhere between 45 and 80 per cent of the trunk’s load. The exponent alone does not choose between them; where along the crown it changes does.

Other crowns #

The pattern holds away from a planar crown, with the numbers the closed forms give. At the drawn trees’ length ratio of 0.74 the weight’s limit is 1.151 and a fifteen-generation trunk reads 1.134. For a crown filling a volume the limit is 1.500 and the trunk reads 1.407, lagging further because its series ratio, $2\lambda^5 = 0.63$, is nearer critical.

With lengths halving, every load’s series converges fast and the measured trunk sits on its limit: 1.500 under tips, 0.999 under wind on the wood, 0.500 under weight. That crown is the one case where three loads give three exponents exactly, a factor of three apart from top to bottom.

Lengths halving leave no room for three #

The crown with lengths halving puts a bound on every mechanical reading at once. Its three loads give 1.5 on the tips, 1.0 as wind on the wood and 0.5 as weight, so a crown shortening that fast and sized by any mix of them conserves an exponent between a half and one and a half, at every junction deep enough to have settled. Murray’s flow rule gives three there as everywhere.

That makes such a crown a sharp instrument in one direction. A fitted exponent near three on a crown whose lengths halve cannot come from any of the three loads, and it cannot come from measurement error either, because an error on the radii pulls a fitted exponent down at every level measured, never up. A reading of three there is evidence for flow sizing against all three mechanical rules together, and a reading of one is evidence for the wood carrying the load.

The drawn trees under their own weight #

The branching trees drawn beside the cube law shrink each branch to 0.74 of its parent and set their widths from Murray’s exponent of three, at an angle nothing derived. At that length ratio a stress rule gives 2.091 under a load on the tips, 2.091 as the limit under wind on the wood — which a fifteen-generation trunk reaches only as 1.912 — and 1.151 under the weight of the wood, which a fifteen-generation trunk reads as 1.134.

So those drawings are drawn at an exponent none of the three mechanical loads gives at their own length ratio, and nearly three times the one the wood’s weight gives. That is not a defect in them, since they illustrate the cube law. It is a measure of how far apart the two families of rule are on one geometry: at 0.74 the flow rule and the weight rule differ by 1.85 in exponent, and a single even fork already tells an exponent of three from one of two.

A parameter a crown could contradict #

The handover from the leaves’ two to the wood’s one is governed by a single quantity, the share of the trunk’s bending moment the wood carries. That share is computed from two loads a person could estimate on a real crown — the wind on its leaves and the weight of its wood — and it predicts a whole profile of exponents, not a single number.

That is the kind of parameter a mechanism has to supply: one that corresponds to something measurable and that the tree could disagree with. A crown whose wood carries most of its trunk’s moment but whose trunk conserves two contradicts stress sizing under the worst-case arm, and one whose exponent is highest near its twigs and lowest at its trunk is doing what this account predicts. Neither measurement is made here.

What the arm assumes #

Every number above rests on the perpendicular arm. For a load on the tips it idealised the wind, and for wind on the wood it does the same. For weight it overstates the moment at every branch that is not horizontal and sets the net moment on a vertical trunk of a symmetric crown to something large where the real value is nothing.

So the weight-sized tree here is a tree sized for the worst gravitational moment each branch could see, as if its crown were turned in every direction at once. A tree that leans carries a net moment and a symmetric one carries none; neither is this tree, and computing either needs a crown with a real orientation, which none of these has.

What this does not establish #

That any tree is sized by stress, that its crown is self-similar, or that its load is shared between leaves and wood in any particular proportion. Nothing here measures a tree. It does not say Da Vinci’s rule is mechanical or that Murray’s is not: a trunk carries sap as well as load, and the cube law’s own derivation is untouched by any of it.

What it does establish is narrower. The stress rule’s two at a planar crown needs its load at the tips; carried by the wood’s weight the same rule gives one, and carried by wind on the wood it gives two only in an infinitely deep crown.

What would withdraw it #

A fixed point that fails to converge, or converges to a different tree from a different start. A tree with no load on its wood that differs from the lever-arm tree. A straight tree under its own weight whose interior departs from $1/(2\log_2(1/\lambda))$. A planar crown whose trunk exponent under weight passes one, or under wind on its wood stops climbing short of 1.85 at fifteen generations. Each of these is checked every time the measurement runs.

Still open: a crown sized for how far it bends #

Stress is one criterion; stiffness is another. McMahon’s elastic similarity sizes a branch so that its own weight deflects it by a fixed share of its length, which for a single beam gives radius proportional to length to the three halves rather than the square. The next test is a crown sized that way under its own weight and under loads on its tips: whether it conserves a single exponent at all, and which length ratio would make that exponent two.