A base that gives

A crown that would rather not buckle sized every branch as a column fixed at its base and free at its top, held a fixed share below its buckling load, and found three things: with every load along its branches the crown conserves the deflection rule’s deep exponent at every junction; under gravity each junction conserves a closed form in its length ratio, its fork and its parent’s tilt; and joined with bending the crown divides by direction into an upright core sized against buckling and a spreading shell sized for bending, with a boundary that is a cone of directions rather than a level in the crown.

Every one of those findings stood on one assumption, and the essay named it last. A branch’s base is its parent’s tip, and the parent bends under the same load it passes on. A column whose base turns buckles sooner than a clamped one, and how far the base turns depends on the parent’s stiffness — which is to say on the parent’s radius, the very thing the rule is solving for. The measurement here is the same crowns with bases that give.

A column on a spring #

A column of bending stiffness $EI$ and length $L$, free at its top and held at its base by a rotational spring of stiffness $k$, buckles when its axial load reaches $EI\varphi^2/L^2$, where $\varphi$ is the root between nought and a right angle of

$$\varphi \tan \varphi = K, \qquad K = \frac{kL}{EI}.$$

On a clamped base $K$ is unlimited, $\varphi = \pi/2$, and the load is Euler’s $\pi^2 EI/(4L^2)$. So a column on a spring keeps $(2\varphi/\pi)^2$ of the load it would carry clamped — its margin, in what follows.

The two ends of that curve say different things. On a stiff base the margin creeps up to one. On a soft base $\varphi^2 \to K$ and the load falls to $k/L$, which is the load at which a perfectly rigid rod standing on that spring tips over. It does not contain $EI$. A column on a soft base cannot be saved by being made stiffer: stiffen it without limit and its buckling load rises only to $k/L$, a ceiling set by the base alone. That ceiling decides most of what follows.

What a parent gives #

The base of a branch is its parent’s tip. A moment there turns the tip of a cantilever by $M L_p/(E I_p)$, so the parent alone offers a rotational stiffness of $E I_p/L_p$. The parent’s own base turns too, and a moment applied at its tip reaches that base undiminished, so counting the whole path to the ground, the compliances add: $\sum L_j/(E I_j)$ over every ancestor. The ground is taken as rigid. Both bases are read here: the parent’s, and the whole path’s.

On the crown sized as if clamped, every quantity in $K$ is known in closed form. A branch’s $r^4$ goes as its length squared times its compression, so a daughter at tilt $\theta_d$ on a parent at $\theta_p$, with $\lambda$ its length over its parent’s, has

$$K = \frac{2}{\lambda} \cdot \frac{\cos\theta_p}{\cos\theta_d}$$

on the parent’s base — and $K = 2/\lambda$ when every load runs along its branch. The crowns are the ones the clamped essay measured: thirteen generations at the planar ratio $\lambda = 2^{-1/2}$, forks turned 15°, 20° or 30°, equal loads on the tips.

Fifty-six per cent, everywhere #

With every load along its branch, $K = 2/\lambda = 2.828$, $\varphi = 1.1766$, and every branch of the crown but the trunk keeps 0.561 of the load it was sized to carry. Not roughly: every one of the 8,190 branches above the trunk sits on that number to within $10^{-9}$. The trunk stands on the ground and keeps all of its load.

A uniform shortfall is the mild kind. $K$ is a ratio of radii to the fourth power, so it does not change when every radius is scaled alike, and a crown thickened by $1/0.561$ in $r^4$ — 1.155 in radius — carries every load at exactly the share of its real buckling load that the clamped rule intended. The exponent its junctions conserve is untouched by the scaling and stays at two. So on an axially loaded crown the clamped rule is wrong by a constant, the constant has a closed form, and the correction is a factor on every branch that the exponent cannot see. The trunk, which needed no correction, is then over-built by the same factor.

A depth effect from the ground #

Counting the whole path to the ground changes that, and it changes it in a way the clamped crown could not show.

The first generation above the trunk keeps 0.561 on either base, since its only ancestor is the trunk, standing on the ground. The second keeps 0.482, the third 0.458, the fourth 0.451, and by the sixth the margin has settled at 0.447 and stays there to the tips. Each ancestor adds its compliance, and the compliances fall off geometrically toward the ground, because a branch’s $L/r^4$ goes as one over its length times its compression and so grows by a factor of $2/\lambda$, about 2.8, with every generation outward. The sum converges within a few terms, to $1/(1 - \lambda/2)$ of the parent’s own share, which puts the deep crown at $K = 1.828$.

The clamped buckling crown had no depth effect at either end: its compression is a tip count, exact from the outermost junction to the trunk. The crown sized for how far it bends had one near the tips, where the arms have not converged. The giving base adds one near the ground, where a branch’s path is short. The outer generations, which are the ones a field measurement can reach, read the settled value, so this one costs a survey nothing.

Under gravity, a closed form in the turn #

Gravity makes $K$ depend on direction, through $\cos\theta_p/\cos\theta_d$.

Every branch of the gravity-loaded crowns at all three fork angles keeps the margin that closed form gives, to $10^{-9}$. With forks turned 20° the trunk’s two daughters keep 0.577. From a parent at 20° a daughter turning outward to 40° keeps 0.613 and one turning back to upright 0.545; from a parent at 40°, 0.667 and 0.507; from 60°, 0.795 and 0.449; from 80°, the daughter turning back to 60° keeps 0.296.

The fork’s own angle sets how wide that range is, and the fork angle is not free: the angle the cost chooses predicts it from the same radii, so on a real crown the margin and the fork are two readings of one sizing.

The asymmetry is the whole story. A daughter turning outward points further from vertical than its parent, so it carries less of the weight as compression than its parent does and stands on a base that is stiff for its load. A daughter turning back toward vertical carries more of the weight as compression than its tilted parent, and the parent was sized for its own smaller compression: the base is soft for the load it now holds. From every parent tilt the daughter turning back keeps less than the one turning out.

Where a crown is weakest #

Drawn, the weak branches are where the argument puts them: on the flanks, where the crown has turned furthest and every fork sends one daughter back up. With forks of 20° the margins run from 0.296 to 0.795; with forks of 15°, from 0.390 to 0.719; with forks of 30°, from 0.417 to 0.695. The widest spread is at the middle fork angle, because at 30° a parent past 60° has daughters at or beyond level, which carry no compression and are not counted, and the steepest back-turn left is from 60°.

Counting the whole path moves every margin down and keeps the ordering: under gravity at 20° the least any branch keeps is 0.245, in the ninth and eleventh generations, turning back from 80° to 60°.

No single factor restores a gravity crown #

On the axial crown one factor fixed everything. Under gravity it cannot. Margins from 0.296 to 0.795 in one crown mean a uniform thickening either leaves the back-turned daughters short or builds everything else up to 3.4 times the $r^4$ it needs. The correction has to be made branch by branch, and a branch’s correction changes its daughters’ bases. So the natural rule is the one the question asked for: size every branch, from the trunk outward, so that it carries the same share of its own buckling load on the base its parent actually gives — $r^4 = \kappa(\pi^2/4)NL^2/\varphi^2$, solved with $\varphi$ depending on $r$ through $K$.

Sized for the base it has, a crown alternates #

On the axial crown the rule does something no clamped crown did. The trunk was sized for the ground, so its daughters stand on a base that is soft for them, and to carry their share they must be 1.134 times as thick as the trunk itself. That first fork conserves nothing: no exponent makes two branches each thicker than their parent add to it. Those thick daughters are then a stiff base for their own daughters, which can be thin; thin branches are a soft base for the next generation, which must be thick. The junctions alternate about two — 0.878 at the first fork above the trunk, 17.85 at the next, then 1.100, 5.353, 1.275, 3.670 — and the swing shrinks slowly, to 2.660 and 1.640 at the tenth and eleventh.

On the whole path’s base the same alternation dies within a few generations: 0.998, 3.248, 1.619, 2.273, 1.872, and within a hundredth of two from the tenth junction on. Each branch’s base now includes every ancestor’s give, which averages the thick and thin generations below it and damps the swing.

The swing has not finished by the tips. At the outermost branches the parent’s-base crown carries 1.568 times the clamped crown’s $r^4$, where the uniform thickening that restores the axial crown would give 1.782: thirteen generations are not enough for a zigzag damping this slowly to reach the crown it is heading for.

What a fit across the crown reads #

A survey does not read one junction; it fits one exponent to many, by least squares on the sum of the daughters’ powers. Pooled over every junction from the first generation to the eleventh, the resized crown on its parents’ bases reads 1.761. Over the outer four generations, where a survey’s junctions actually are, 1.784. Neither is two, and neither is anywhere near the alternating values themselves. The outermost generation carries half of all the junctions and sits on the low side of the swing, at 1.640, so a pooled fit is pulled toward it: it reports a crown sized by a rule that conserves about 1.76, which no junction in the crown conserves. On the whole path’s base the same fits read 1.997 and 2.000. That is the shape a sample that is confidently wrong described for a crown whose junctions differ by generation — a fit with a small error bar about a number that belongs to no junction.

Neither is a small effect, and neither is what the question expected. It asked whether the rule for a branch, once it depends on the branch below, would still give a crown; it gives one whose radii zigzag from generation to generation, with a ratio of daughter to parent that swings from 0.45 to 1.13 near the ground and settles on the clamped crown’s 0.707 only slowly, and only on the whole path’s base.

A daughter no radius can hold #

Under gravity the resized crown meets the ceiling. A branch whose load times its length exceeds what its base could hold as a rigid rod cannot be sized by any radius at all.

With forks turned 20°, 226 branches of a thirteen-generation crown are like that, the first in the fifth generation, and every one of them turns back toward vertical: 112 from 80° to 60°, 104 from 60° to 40° and 10 from 40° to 20°. The 3,904 branches above them are left without a size, since nothing can be said of a branch that stands on a base the rule cannot make. With forks of 15°, 232; with forks of 30°, on the whole path’s base, 62.

This is the ceiling from the spring, met in a crown. A daughter turning back carries more compression than its parent’s tilt allotted, and when the product of that compression and its length passes its parent’s end stiffness, making the daughter stiffer does nothing. The requirement belongs to the parent. A rule that sizes each branch alone, for the load above it and the base below it, cannot express that, and the branches where it fails are exactly the ones the closed form had already marked as weakest.

The joined crown, again #

The clamped essay’s most visible finding was the division by direction. Joined with bending — each branch taking the larger radius of the two criteria, balanced on the trunk at a stated tilt — the deep crown was buckling-sized up to some tilt and bending-sized beyond it, whatever the generation. Resized for bases that give, that division no longer holds everywhere.

On clamped bases, balanced at 30°, the deep crown is buckling-sized to 20° and bending-sized from 40°. On the parent’s base the buckling core reaches 60° while bending starts at 40°: branches at 40° and 60° deep in the crown are sized by one criterion here and the other there, and 212 branches have no radius. Balanced at 50° on the parent’s base the two meet at 60°, with 346 unsizable. Balanced at 50° on the whole path’s base, or at 70° on the parent’s, the cone survives, buckling to 60° and bending from 80°, but with 436 and 290 branches the rule cannot size.

So the core does not narrow; where the division holds at all, it widens. A soft base makes buckling the larger requirement at tilts where bending used to be, which pushes the core outward, and the core’s own edge becomes ragged because a branch’s buckling requirement now depends on its parent’s radius as well as its own direction. The junctions go with it. Balanced at 50°, the buckling core’s junctions read 0.517 to 20.6 on the parent’s base, where on clamped bases they read 1.881 to 1.914: they have left any closed form in the tilt, because each one now carries its grandparent’s give as well as its parent’s.

The three answers #

The question asked whether the upright core narrows, whether its junctions stay on a closed form in the tilt, and whether the boundary stays a cone. Read on the crown sized as if clamped, the answer is a closed form of a new kind — every branch’s margin, in its parent’s tilt and its own — and a constant factor on an axial crown. Read on the crown resized for its bases, the core widens rather than narrows, its junctions leave the closed form, and the cone holds at some balance angles and not others, with hundreds of branches the rule cannot size at all.

The two readings differ because the second rule asks each branch to repair its own margin, and the margin is not the branch’s to repair. That is the same shape of result three rules, one exponent and a spread of branch lengths found from other directions: an exponent measured on a crown is a statement about the rule that sized it, and a rule written for one branch at a time misdescribes a crown whose branches hold each other up.

What a field measurement would have to record #

The clamped essay asked for the tilt of every branch beside every junction. A giving base asks for the turn at every fork — whether each daughter leans out from its parent or back toward vertical — since that alone sets the margin on the clamped crown. It asks, harder, for the stiffness of the joints, which a branch radius does not give: a fork with reaction wood or a swelling at its base is a stiffer base than its parent’s radius implies, and a swelling at the fork has already shown what such a swelling does to the exponent read across it.

What this does not establish #

That a real junction is a rotational spring and nothing else. A parent also deflects sideways under a daughter’s load, and the two daughters of a fork load one tip together; the buckling mode of a whole crown couples every branch to every other, and a branch-by-branch criterion with a spring at its base is the simplest honest approximation of that, not the thing itself. That the crown keeps its shape while its radii change. That the ground is rigid — a trunk on roots gives too, which would add a compliance at the bottom of every path. And, as before, that trees are sized against buckling at all.

Findings that would overturn it #

A branch of the clamped-sized crown whose margin on its parent’s base departs from $(2\varphi/\pi)^2$ with $\varphi \tan\varphi = (2/\lambda)\cos\theta_p/\cos\theta_d$. An axially loaded crown on the parent’s base whose margins differ from branch to branch. A resized axial crown whose junctions do not alternate about two. An unsizable branch that turns outward. Each would mean the arithmetic here is wrong rather than incomplete.

Still open: a parent sized for the base its daughters need #

Every unsizable branch says the same thing: the requirement was its parent’s. A parent standing under a daughter that carries compression $N_d$ over a length $L_d$ must give it a base stiffer than $N_d L_d$, so its own $r^4$ must exceed a constant times $L_p L_d N_d$ — a term from the daughter’s load and both lengths. That is again a load times a length squared, the power law buckling and bending already share. The next measurement is a crown sized by the larger of each branch’s column term and the base term its daughters demand, with a stated margin on the second: whether it conserves a closed form in the turn at each fork, whether the alternation disappears once a parent is sized for its daughters rather than a daughter for its parent, and whether the cone of directions comes back.