Moving the wood saves almost nothing

A parent sized for its daughters gave every branch of a crown the radius it needed to carry its load on the base its parent provides, and found the cheapest crown in wood at a margin of about 1.7 over the rigid-rod ceiling. A crown sized branch by branch buckles as one then assembled that crown as one frame of beam-column elements and asked for its lowest buckling load. The answer was 0.205 of the load it had been sized for on the seven-generation crown. Every branch had been safe on its own and the whole was not.

That essay, taking up the case of a crown that would rather not buckle, repaired the crown by the obvious route: multiply every $r^4$ by the same factor until the frame just holds, which costs a factor of $1/\sqrt\lambda$ in wood. But the same essay had split the mode’s strain energy by branch and found it concentrated at the bottom. Its closing section therefore made a suggestion that sounds like plain engineering sense. The mode lives in the trunk and the first forks, thickening a twig costs wood the mode hardly uses, so thicken each branch in proportion to its share of the mode’s energy instead. It asked how much wood that would save, what would become of the exponents, and whether the cheapest margin would move again.

The saving is small enough to be the main result. And the reason it is small is the most useful thing in the measurement.

Moving wood towards the mode #

The redesign keeps the crown’s shape and its compressions, and changes radii only. A branch’s useful measure is its share of the mode’s strain energy over its share of the total wood. At an optimum for a buckling load, this ratio is the same for every branch, which is the ordinary optimality condition for a structure whose stiffness goes as the square of its material: if one branch’s energy per unit of wood were higher than another’s, wood moved to the first would raise the load.

So each step multiplies a branch’s wood by that ratio raised to a quarter, which damps the change, and rescales so that total wood is unchanged. The frame is re-solved, since a new mode may have taken over from the old one, and the crown’s wood is read as what it would need to hold its loads exactly: its wood over $\sqrt\lambda$. Twenty-four steps are run. Nothing prevents a branch getting thinner, down to a quarter of what the sizing gave it. A version in which no branch may go below the size the per-branch model gave it is read as well, and its answers are compared below.

Four starting margins are used, 1.2, 1.7, 2.5 and 4, at three depths of 31, 63 and 127 branches. The frame is otherwise the one the earlier essay built: each branch cut into four elements, rigidly joined, the trunk clamped.

What the crown does #

The first thing the dial shows is how little happens. From the crown thickened alike, the first step has already done most of what will ever happen, and by the second no change can be seen by eye. The trunk loses a very little and the twigs lose more, with the third generation gaining a little. A person shown the first picture and the last and asked which is which would need to measure them.

That is, numerically, what is happening too. From the margin of 1.7 on the sixty-three-branch crown, wood to hold the loads is 44.89 thickened alike, 43.81 after one step, 43.77 after two and 43.76 after four, where it stays. The reduction, in all, is 2.5 per cent.

How much is saved #

The saving depends strongly on the starting margin, and only at the bottom of the range. At 1.2 on the sixty-three-branch crown the uniform repair costs 14.5 per cent over the best, a real saving, since that crown is a zigzag of over-thick and under-thick generations. At 1.7, 2.5 per cent. At 2.5, 1.5. At 4, 1.0. Across the depths the numbers are similar: on thirty-one branches, 14.8, 2.5, 1.3 and 0.5; on a hundred and twenty-seven, 13.3, 2.4, 2.0 and 1.7.

The earlier essay found that a margin near four was the cheapest once the frame was taken into account. So the crown that was already the cheapest uniform one is within 1 per cent of everything this procedure can reach, at the depth read here. A proposal that promised to save the wood being spent on twigs comes down to about that one per cent. That is too small to ask any developmental program to find, and small enough that real wood’s variation in stiffness, which is far larger than this, would drown it.

The same crown from everywhere #

More striking than the saving is what the four runs arrive at. Starting from crowns as different as a margin of 1.2 and a margin of 4, the sixty-three-branch crown lands on 43.76 from all four. The thirty-one-branch crown lands on 25.73 from all four. The hundred-and-twenty-seven-branch crown lands on 70.91 from all four. The radii agree too, generation by generation. So what was a free constant in the sizing, and an awkward one, disappears: the margin affects the way there and not the place.

So the crown’s shape and its compressions have a single best set of radii against its lowest mode, and that is a tidy statement that nothing in the earlier essays could have made. It gets rid of an unease about the earlier sizing, where one number that the model could not fix had decided the wood. The trade is that this is a stated optimisation and not anything a tree can be assumed to do. A crown holding that exact distribution would have had to solve an eigenvalue problem across its whole frame. Nothing here says one has.

Holding each branch above the size the per-branch model gave it changes the answer little. The sixty-three-branch crown at a margin of 1.7 reaches 43.88 under that constraint, against 43.76 without it, and half of its branches, every one in the last generation, sit against the floor. So the twigs want to get thinner than any local reason would allow, and they gain almost nothing from it when they do.

Why it saves so little #

The argument for moving wood turned on a true observation. The mode’s energy does sit at the bottom, 31 per cent of it in the trunk, and only 4 per cent in the twigs of the last generation. The error was to compare that with nothing. What decides whether wood is being wasted is the energy share against the wood share, and the wood falls away up the crown as fast as the energy does.

A crown thickened alike has 31 per cent of its wood in the trunk, 26 in the first fork, 16 in the second, 13, 8 and 6. The best crown has 31, 25, 19, 13, 8 and 4, equal to its energy shares in every generation. Those two lists differ by three points at the most, in the second fork, and two in the twigs. That is the whole of what moving the wood had to work with.

The uniform repair is close to optimum for a reason that is not a coincidence. Sizing every branch for its own column already gives wood in proportion to something very like its load and length. The frame’s mode is a smooth bending of the entire structure, and a smooth bending of a structure sized that way stores its energy in much the same proportion. A crown sized by a local rule turns out to have its wood nearly where a global rule would put it, and the reason that crown is weak has nothing to do with the distribution. It was weak because local sizing leaves all of it too thin by one common factor.

This leaves the earlier finding in place and gives it a clearer form. A branch-by-branch sizing misses the whole crown’s load by about a factor of five on the seven-generation crown, and the fix for that is a uniform thickening, at a cost of about two and a quarter times the wood. Nothing about where to put the wood changes that factor materially.

Two costs of very different size #

It helps to put the saving beside the cost it was meant to reduce. The crown sized branch by branch at a margin of 1.7 has 14.8 of wood at five generations, 22.3 at six and 32.9 at seven. To hold its loads as a frame, thickened alike, it needs 26.4, 44.9 and 72.6. So the whole-crown check multiplies a crown’s wood by 1.78, 2.01 and 2.21, and that factor grows with each generation. Moving the wood then takes back 2.5, 2.5 and 2.4 per cent.

One of those numbers is a hundredfold larger than the other, in the sense that matters. A theory of how much wood a tree needs has to account for the factor of two. Its local sizing misses that by a wide margin, and a base that gives showed where a share of that softness comes from. A theory of where the wood goes has very little left to explain on this evidence, since almost any sensible sizing puts it close to where the frame wants it. Those two theories have often been treated as one, and this measurement separates them cleanly.

The procedure’s behaviour supports the same reading. It settles in about four steps and sits unchanged for fifteen more. Past about step twenty it starts to wander, with the wood drifting up to 44.2 and 46.3 in the last two steps, and that is why the best crown it visited is kept rather than the last. The cause has not been traced here. It may be that near the optimum two modes close to each other in load begin taking turns as the lowest, a well-known feature of buckling optimisation, and then a step aimed at one damages the other. A valley this flat gives a step of fixed size no reason to stop, and it would explain why so small a difference in wood is reached so fast and then lost.

Thicker at the base, thinner at the tips #

The closing question was whether the result still conserves exponents or drifts toward a thicker base and thinner tips. It drifts. In mean radius by generation the moved crown reads 3.69, 2.79, 2.03, 1.42, 0.94 and 0.54. Between one generation and the next those ratios are 1.32, 1.37, 1.43, 1.51 and 1.74, each larger than the one before. A symmetric fork’s exponent is the logarithm of two over the logarithm of that ratio, so from the trunk outward the exponent runs 2.5, 2.2, 1.9, 1.7 and 1.25. So the best crown does not hold one exponent at all.

Against that, a crown sized at a margin of 4 and thickened alike reads 3.89, 2.75, 1.95, 1.38, 0.97 and 0.53, ratios of 1.41 all the way until the last, which is exactly the crown of fixed exponent two that the earlier essays found above a threshold. It is within one per cent of the same wood. So an exponent that conserves itself exactly and an exponent that falls steadily from 2.5 to 1.25 are two crowns that hold the same loads to within a per cent.

A fitted exponent reads that crown as about 1.46, which shows how little such a number should be trusted. It is one fitted number across a crown whose real exponent changes by a factor of two, and it comes out near the value for the twigs because the twigs are half of all the forks. An estimator’s leverage has been a problem on this subject in several earlier essays, and this is the same weakness showing up on a designed crown rather than a measured one. On the moved crown the fit is the same from every margin, 1.48, 1.46 and 1.45 at the three depths. Thickened alike, it varies from 1.22 up to 2.39 with the margin alone. So a crown’s fitted exponent says as much about how it was sized as it does about what it must carry.

What this does to a claim about exponents #

A long line of essays here has asked what a branching exponent means, whether an organ has one at all, how little it can matter, and what different mechanical criteria predict. This reading adds something awkward to that account. Buckling has been offered as the reason for an exponent near two, by way of elastic similarity. Here is a crown optimised against buckling as a whole, whose exponent falls from 2.5 to 1.25, and a uniform crown of exponent two that holds the same loads with the wood within a per cent. Whichever a real tree has, this criterion barely discriminates between them. A mechanical criterion that is nearly flat over a factor of two in the exponent cannot be the thing that pins an exponent, on this evidence.

That should not be overstated. The flatness is a statement about wood at fixed loads and shape. Other considerations could still bear on it, hydraulics being the obvious one, and the junction measurements ask a separate question about how well any of this can be read from a tree. What it does establish is a limit on the buckling argument specifically. Wood efficiency in a buckling crown will not by itself select an exponent to better than a wide band.

What this does not establish #

The crown is a planar symmetric binary tree with its loads along its branches, and its branches are straight. Real wood has a stiffness that varies far more than a per cent across one tree. The lowest mode is what is targeted, and a crown near several modes at once may need a different treatment; here the procedure settled quickly, but no crown has been checked for several modes sharing the lowest load. Bending under its own weight, wind loads and the rest are not in this frame at all. The optimum found is the best the procedure reached from four starts, which agree to four figures, and that is strong evidence of a single optimum and not a proof.

How this could fail #

A crown of another shape for which moving wood saved far more than a few per cent at a margin above its own threshold would mean these crowns are unusually near their optimum, rather than that crowns generally are. Different starting margins landing on different best crowns would remove the claim that the margin decides only the route. A mode energy share that differed widely from the wood share on the best crown would mean the procedure had not converged. And a real crown measured to have a falling exponent with a falling ratio of this form would make this reading much more interesting than the flatness of its wood suggests.

Still open: a crown with something else to carry #

Buckling under the crown’s own axial loads is one load case, and its wood is nearly flat over a large range of crowns. A real crown bends in the wind, whose load grows with its tips’ leaf area and with height, so that its energy lies in a very different place from a buckling mode’s: mostly in bending at the base of each limb. The next measurement adds a lateral load in a stated ratio to the axial one, repeats the moving of wood against the combined energy, and asks whether the optimum is still as flat, whether the fork exponent still falls from trunk to twig or now climbs, and at what ratio of the two loads a crown sized for one stops being within a per cent of a crown sized for the other.