A crown that would rather not buckle

A branch can fail in three ways. It can break, which is what equal bending stress sizes against; it can sag, which is what equal deflection sizes against; and it can buckle, folding sideways under the weight it carries long before either of the others. The first two each have a crown whose junctions conserve a known exponent. The third has had a single column and a question: the crown sized for how far it bends noted that a column held below its buckling height goes as radius to length to the three halves, the same power as elastic similarity, and asked whether that is a coincidence of the single beam or survives into a crown whose every branch carries the compression of everything above it.

It is not a coincidence, and it does not survive in the form the question expected. The two criteria are one power law, and what separates them in a crown is not the arithmetic of the load but its direction.

Two failures, one power law #

A branch treated as a column fixed at its base and free at its top buckles under a load at its top when that load reaches $\pi^2 EI/(4L^2)$, where $E$ is the wood’s stiffness and $I$ the second moment of its cross-section. Under its own weight it buckles when its length reaches Greenhill’s height, the root of $qL^3 = 7.837\,EI$ with $q$ its weight per unit length. Holding either load a fixed share below its limit, with the second moment going as the radius to the fourth, sizes the fourth power of the radius against the length squared times the load.

The deflection rule sizes the same fourth power against a sum of each load’s deflection kernel divided by the branch’s length, and for a load beyond the branch that kernel is the length squared times three arms less one length. So both rules hold the bending stiffness against a load times a length squared. Fitted across lengths from 1 to 11.4, a column under its own weight goes as length to the 1.5000 and a cantilever under its own weight to the 1.5000; under a load at the top, both to the 0.5000.

Why the coincidence is not one #

The three halves that McMahon called elastic similarity was, in his own first argument for trees, a buckling result: tree heights against trunk diameters, read against Greenhill’s height. The deflection version and the buckling version were always going to agree, because each holds the same quantity — a stiffness per unit of load times length squared — at a fixed value, and a fixed value of one quantity gives one power law however the constant in front is chosen.

That is a statement about what is conserved rather than about any number, and it has a consequence worth testing. A crown in which every branch carries its load along its own axis should conserve, under buckling, exactly what the crown sized for deflection conserves: $4/(1 + 2\log_2(1/\lambda))$, where $\lambda$ is the length of a branch over its parent’s.

A crown whose loads all run along its branches #

It does, and it does so more strictly than the deflection crown does.

Sized against buckling, every junction of a thirteen-generation planar crown conserves 2.000 to within $2 \times 10^{-14}$, from the outermost junction to the trunk. Sized for bending, the same crown’s junctions two generations above the tips conserve 1.326, five generations in 1.869, and the trunk 1.995. The limit is the same. Only buckling reaches it everywhere.

No depth effect, and why #

The reason is that a buckling rule reads a branch’s load and nothing about where that load sits. The compression a branch carries under equal loads on the tips is exactly its tip count, which is exact at the outermost junction and exact at the trunk, so the junction’s two daughters stand to their parent in the same ratio at every height.

The deflection rule reads the arms, and near the tips the arms have not converged: a branch two generations above the tips has a much shorter lever behind it, in proportion to its own length, than one twelve generations up. That is the approach from below every mechanical exponent on these crowns shows — Murray’s flow rule is the other exception, since the cube law reads tips and not arms — and a column does not have it. On a real crown with few generations, that is the largest single difference between the two readings.

Gravity divides a load #

Nothing in a tree carries its load purely along its branches. Gravity pulls straight down, and a branch tilted by an angle $\theta$ from vertical carries the weight above it partly along its axis and partly across it: $\cos\theta$ of it as compression, which is what buckles a column, and $\sin\theta$ of it as a sideways load, which is what bends a cantilever.

Under equal loads on the tips the compression in a branch is therefore its tip count times the cosine of its tilt, and a branch sized against buckling has the fourth power of its radius equal to a constant times its length squared times its tip count times $\cos\theta$, exactly and at every generation. The deflection rule, with loads perpendicular to the arms, has no such factor.

An upright junction reads its fork #

Take the junction at the top of a vertical trunk, whose two daughters lean by the fork’s half-angle $h$ either side. Each daughter carries half the tips at $\cos h$ of their weight, so the junction conserves $4/\log_2\bigl(2/(\lambda^2\cos h)\bigr)$, and that is lower than the deflection rule’s exponent by exactly the $\cos h$ inside the logarithm.

On planar crowns with forks turned 10°, 15°, 20°, 25° and 30°, the upright buckling junction conserves 1.978, 1.951, 1.914, 1.867 and 1.812, each equal to the closed form to the last digit. The same trunks sized for bending read 1.993 to 1.997 — within a hundredth of two at every angle — because a distance does not care which way a branch leans. So a buckling crown’s exponent reads its forks, and a bending crown’s reads its lengths. The fork’s angle has had a prediction of its own, the angle a cost chooses, and on a tree sized by stress the angle and the exponent were two readings of one sizing; under buckling they are no longer independent at all.

A tilted parent reads something else #

A junction whose parent is itself tilted has daughters at $\theta - h$ and $\theta + h$, carrying different shares of their weight as compression, and it conserves the exponent that solves $\sum \bigl(\lambda^2\cos(\theta \pm h)/(2\cos\theta)\bigr)^{p/4} = 1$. That is a closed form in three numbers: the length ratio, the fork’s half-angle and the parent’s own tilt.

With forks turned 20° an upright parent’s junction conserves 1.914 and a parent tilted 60° conserves 1.751; past 70° a daughter reaches level and nothing is conserved. With forks turned 30° the two readings are 1.812 at an upright parent and 1.774 at one tilted 30°, and a daughter reaches level past 60°. Every junction of every gravity-loaded crown measured sits on its line to the last digit, and refuses exactly where the line ends.

The exponent belongs to the path #

A branch’s tilt is the sum of the turns on its path from the ground: right, right, left, right. So two junctions in the same generation, at the same height and with the same number of tips above them, conserve different exponents if their paths turned differently, and two junctions generations apart conserve the same exponent if their paths summed to the same tilt.

That is the answer to the question as asked. The compression a branch carries does depend on its whole path to the ground, and the dependence is entirely through its direction. A buckling crown does not conserve one exponent; it conserves one exponent per direction, and the exponent is exact.

Where buckling stops sizing anything #

The cosine has a sign. A branch at level carries no compression, and one pointing below level is in tension, so a buckling rule assigns it no radius at all. On a symmetric crown the share of a generation that points level or down is a binomial count of its paths’ turns.

With forks turned 30° the share is a quarter of generation three, 45.3 per cent of generation seven and 54.8 per cent of generation eleven. With forks turned 20° it is none of generation three, 12.5 per cent of seven and 22.7 per cent of eleven; at 15°, 1.6 and 6.5 per cent. On the crowns measured, the branches left unsized are exactly these, generation by generation.

A criterion that cannot size a whole crown #

So buckling on its own is not a sizing rule for a spreading crown. It sizes an upright core and says nothing about most of the outer generations, and a branch it does not size is not a branch with no requirement; it is a branch whose requirement comes from somewhere else. The obvious somewhere else is bending, which is indifferent to direction, and the obvious joint rule is the one an engineer would write: each branch takes whichever radius is larger.

That joint rule has a free constant. The two criteria put different constants in front of the same power law, and their ratio is set by how heavy the leaves are against how hard the wind pushes on them, and by how close to buckling and how far out of true each allowance lets a branch go. Nothing in the arithmetic fixes it. It is carried here as the tilt at which the two criteria would balance on the trunk.

A crown split by direction #

The upright branches are sized against buckling and the spreading ones for bending, and the picture already shows what the measurement confirms: the split follows direction. In a thirteen-generation crown of that shape buckling sizes 80 per cent of all branches, most of them near the top, and bending sizes the rest, most of them near the tips, because that is where the branches that have turned furthest are.

The choice is made by direction #

Seven or more generations above the tips, the widest branch buckling sizes tilts less than the narrowest branch bending sizes, at every one of five balance angles and whatever generation each branch is in. Balanced at 30° the deep crown is buckling-sized to 20° and bending-sized from 40°; balanced at 50°, to 40° and from 60°; balanced at 70°, to 60° and from 80°. The upright core widens as the balance angle does, in steps of the fork’s angle.

Why direction and not height #

The deep-crown bending term is proportional to a branch’s length squared times its tip count, exactly as the buckling term is, so the ratio of the two is a constant times the cosine of the tilt. Height cancels. A branch governed by buckling at generation three would be governed by buckling at generation nine if it pointed the same way, and the boundary between the two regions is a cone of directions rather than a level in the crown.

That is a checkable claim rather than an interpretation, and it is the one the boundary figure checks: a boundary set by height would show the two bars overlapping at some generation, and deep in the crown they never do.

Near the tips the core widens #

The cancellation needs converged arms, and near the tips the arms are short. There the bending term is smaller than its deep value, so buckling wins at tilts it would lose further in. Within six generations of the tips, buckling sizes branches out to 60° at every balance angle measured, including the crown balanced at 30° whose deep core stops at 20°.

This is the depth effect of the deflection rule showing up in a new place. It makes the outermost generations the least representative of the crown’s own division, which is the opposite of what a measurement needs, since the accessible junctions are the outer ones.

Junctions across the boundary #

Inside each region a junction conserves what its own rule gives it. Across the boundary the parent is sized by one rule and a daughter by the other, and that junction is different.

At a balance angle of 50° the buckling junctions read 1.881 to 1.914 and the bending junctions 1.869 to 1.976, while the straddling junctions reach 2.189. At 70°, where the core is widest and the boundary runs through the most steeply tilted forks, the straddling junctions reach 2.457. A planar crown sized by either rule alone conserves two.

Why a straddling junction reads high #

A branch sized by the larger of two requirements is at least as thick as either would make it. Inside a region the larger requirement is the same one for the parent and both daughters, and the junction reads that rule. At the boundary a daughter’s requirement comes from the rule its parent was not sized by, and it is larger there than the parent’s rule would have given it — that is what being across the boundary means. So the daughter is relatively thick, and a relatively thick daughter needs a higher exponent to conserve anything.

The effect grows with the balance angle because the boundary moves out to more tilted forks, where the cosine changes fastest between a fork’s two daughters.

What a field measurement would have to record #

The exponent, the length ratio and the depth, which three rules, one exponent already asked for, and now the tilt of every branch measured. Without the tilts, a buckling crown’s exponents look like scatter; with them, each junction has a closed form and the measured exponent either sits on it or does not. That is a sharper version of the warning fitting the exponent gives about pooling junctions of different kinds into one fit.

It also asks for something harder, which is where in the crown the criterion changes hands. A spread of branch lengths showed that a regression on the right quantity can name a rule where the exponent cannot; here the right quantity is the parent’s tilt, and a junction’s exponent regressed on its parent’s cosine would separate an upright core sized against buckling from a shell that ignores direction. That regression has not been run on these crowns and is named rather than claimed.

What this does not establish #

It does not establish that a branch is a column with a rigid base. Its parent bends under the same load, so the base rotates, and a column whose base rotates buckles at a lower load than one whose base is clamped. It does not establish that leaves load a crown at its tips alone, that any ratio of leaf weight to wind is typical, or that trees are sized against buckling at all rather than for the height at which they would buckle as whole trunks, which is McMahon’s original observation and a different claim.

Nor does it say anything about wood: every load here is on the tips. A crown that carries its own wood showed that moving the load onto the branches changes the exponent a stress rule conserves from two to one, and the buckling version of that crown is a fixed point in which a tilted branch’s own weight divides by its own cosine.

The claim, reduced #

A column held below its buckling load and a cantilever held to a fixed deflection are one power law, so a crown whose loads all run along its branches conserves $4/(1 + 2\log_2(1/\lambda))$ under either — at every junction under buckling, and only deep in the crown under bending. Under gravity a buckling junction conserves a closed form in the length ratio, the fork’s half-angle and its parent’s tilt, branches at or below level are not sized, and a crown sized by the larger of buckling and bending divides by direction, not height, with its boundary junctions conserving more than either rule.

What would withdraw it #

A column and a cantilever with different similarity exponents under the same load. An axially loaded buckling crown departing from the deflection rule’s deep exponent at any junction. A gravity-loaded buckling junction off its closed form. A joined crown in which, deep in the crown, a branch at one tilt is sized by buckling at one height and by bending at another. A straddling junction that reads below both regions it joins. Each is checked whenever the crowns are measured.

Still open: a base that gives #

Every branch here is a column clamped to a parent that does not move. A real parent bends under the same load it passes on, so the daughter’s base rotates, and a column on a rotating base buckles at a lower load than a clamped one — with a flexible enough base, at an arbitrarily low one. Worse, how far the base rotates depends on the parent’s own stiffness, and so on the parent’s radius, which is what the rule is solving for.

The measurement is the joined crown again with each branch’s buckling load taken for a base whose rotational stiffness is its parent’s bending stiffness over its parent’s length: whether the upright core narrows, whether its junctions still sit on a closed form in the tilt, and whether the boundary stays a cone of directions once the rule for a branch depends on the branch below it as well as on the load above it.