The lengths that name the rule

Every crown in the essays on sizing rules so far has had one length ratio. A branch is $\lambda$ of its parent’s length at every fork, both daughters alike, and under that assumption three rules give three exponents: Murray’s flow rule conserves three, equal bending stress conserves $3/(1+\ell)$ and equal deflection $4/(1+2\ell)$, with $\ell = \log_2(1/\lambda)$. An exponent names the rule only with a length ratio beside it, and at the crown that fills a plane it names nothing, because both mechanical rules give exactly two there.

No tree has one length ratio. Two daughters of one fork are rarely the same length, and neither is a fixed fraction of its parent. So the question that essay left is what a spread does: whether each rule’s junctions still conserve one exponent, whether it is the one the mean ratio predicts, and whether the spread destroys the identification or supplies a new one.

A crown whose forks do not agree #

The crowns here are the same crowns, with one change. At every fork each daughter’s length is its parent’s times the mean ratio times a factor drawn independently from a stated range in log length, from a stated seed so that the same crown is drawn every time. At a half-width of 0.3 the factor runs from 0.74 to 1.35, so one daughter can be nearly twice as long as its sibling.

The picture on the right looks more like a tree and is harder to describe. Each of the three rules is then applied to it exactly as before: flow sizes the cube of a radius against the tips a branch feeds, stress against the sum of the tips’ lever arms, and stiffness sizes the fourth power against the sum of the arms’ deflection kernels divided by the branch’s own length. The loads are the same equal loads on the tips throughout.

Flow does not notice #

The first result is exact. Every junction of every crown sized by Murray’s rule conserves three, whatever the spread and whatever the mean ratio, to the last digit a double holds. Flow reads how many tips a branch feeds — the whole content of the cube law is that the length cancels — the crowns are symmetric in how they branch, and a daughter’s length changes nothing about how many tips lie above it.

That was already the reason the flow rule is a horizontal line against the length ratio, and a spread is only a length ratio that varies. It is worth stating anyway, because it means a spread of lengths is a test with a null answer built in: whatever a crown’s junction exponents do as the lengths vary, a flow-sized crown’s do nothing at all.

The mean survives #

The two mechanical rules are not indifferent, and the obvious worry is that a spread shifts their exponents so far that the length ratio no longer predicts them. It does not. Measured on thirteen-generation crowns at the planar ratio, the mean exponent of an interior generation rises by 0.013 under equal stress and 0.050 under equal bending at a half-width of 0.3, and by 0.002 and 0.007 at a half-width of 0.1.

Those numbers grow as the square of the spread: a third of the width gives about a sixth of the rise. So the exponent the mean length ratio predicts is still the exponent a spread crown conserves on average, to within a few hundredths over any spread a real crown is likely to have. The upward drift is real and it has a sign, and it is small beside everything else a spread does.

What the spread does instead #

It scatters the junctions, and a width is a quantity with information in it that a mean discards, which is the argument the second moment is the measurement makes about the side counts of cells. In a crown with one ratio every junction of a generation conserves the same exponent, because every one of them is the same fork scaled. In a crown with a spread each junction has its own pair of daughters, and each pair gives its own exponent.

Eight generations above the tips the stress-sized junctions read $1.977 \pm 0.085$ and the bending-sized ones $2.008 \pm 0.227$, and the widest bending junction in that generation conserves 2.593 while the narrowest conserves 1.573. The means fall away towards the tips exactly as they do in a crown with one ratio, which is the depth effect a crown sized for how far it bends measured. The boxes keep their width all the way up.

A trunk is a sample of one #

That last point is the practical one, because the junction most often measured on a real tree is the lowest. On a crown with one ratio the trunk is as good as any junction. On a crown with a spread it is one draw from a distribution.

With no spread all forty trunks read 1.997 under both rules. At a half-width of 0.2 the stress trunks run from 1.884 to 2.112 and the bending trunks from 1.733 to 2.376; at 0.4, from 1.783 to 2.239 and from 1.533 to 2.931. A single bending-sized crown can therefore present a trunk reading close to Murray’s three while its interior conserves two. Nothing about that crown is unusual. It is one draw.

The obvious prediction is three times too wide #

The natural way to predict the scatter reads a junction off its own daughters. Deep in a crown a branch’s moment is its tip count times its length times a constant, so a daughter’s radius relative to its parent’s depends on that daughter’s length alone, and the junction’s exponent solves an equation in the two daughters’ length ratios.

That local form is straight in the spread, which is right, and it is far too steep. At a half-width of 0.4 straight-armed crowns scatter by 0.105 under stress and 0.301 under bending; the local form predicts 0.343 and 0.587. It is 3.3 times too wide under stress and twice too wide under bending.

Why: a long daughter thickens its parent too #

The mistake is in the parent. Its moment is not its own length times a constant; it is its own length plus the mean of its two daughters’ arms. So a long daughter does two things at once. It thickens itself, and it thickens its parent, and the second partly undoes what the first does to the junction.

Carrying that one step up gives a two-level form, in which each daughter’s share of its parent’s radius depends on both daughters’ lengths. It predicts 0.091 and 0.316 at a half-width of 0.4 against the crowns’ 0.105 and 0.301, and it holds as closely at every spread measured. The form also predicts the rise in the mean, with the right curvature and too large — 0.035 and 0.100 against the crowns’ 0.020 and 0.085 — because a parent’s arm carries its grandchildren’s lengths as well, and the form stops one level short.

Stiffness scatters more, everywhere #

Read against the whole family of length ratios, the scatter per unit of spread behaves differently under the two rules. Under stress it barely depends on the mean ratio at all, sitting between 0.42 and 0.49 per unit standard deviation of log length from halving lengths to 0.9. Under bending it climbs steeply, from 0.66 at halving lengths to 2.44 at 0.9.

The reason is that stiffness reads a length squared where equal stress reads it once, so any one daughter’s length counts for more, and as the mean ratio approaches one the arms converge more slowly and every length in a subtree counts. At the planar crown, where the two rules conserve the same exponent, the bending scatter is 2.62 times the stress scatter. That looks like the separation three rules could not find.

And a caliper error erases it #

It is not, and a real measurement is the reason. Every radius read off a real branch carries error, and an error in a radius moves the exponent fitted through it. Applied here, junction by junction, a one per cent error in every radius on a crown with no spread at all scatters its stress junctions by 0.072 — more than four fifths of the 0.086 that the whole spread of lengths produces.

The two sources add in quadrature, as independent ones should: with both present the stress scatter is 0.108 at a one per cent error, 0.159 at two and 0.416 at five, where it has grown to nearly five times its exact value. A stress-sized crown measured with two per cent error scatters more than a bending-sized crown measured exactly. The scatter confounds the rule with the calipers, and nobody measuring a crown knows the second well enough to take it off.

The measurement that separates them #

Separating them needs something the radius error cannot reach, and the spread supplies it. The error in a radius is independent of how long the branches are. The effect of the spread is not: a junction whose daughters are both long reads a higher exponent than one whose daughters are both short, and that dependence is a relation between two measured quantities rather than a width.

Regressing each junction’s exponent on the sum of its two daughters’ log length ratios gives a slope. Measured exactly on one crown at the planar ratio, the stress exponents follow a slope of 0.298 and the bending exponents 0.858. With every radius two per cent wrong the clouds swell — the share of the stress scatter the lengths explain falls from 62 per cent to 20 — and the slopes read 0.288 and 0.866.

Why the sum and not the difference #

The regression uses the sum of the two daughters’ log length ratios, and that choice is the linearisation rather than a convenience. Expanding the junction’s equation about the symmetric crown, a small change in either daughter’s log length moves the exponent by the same amount at first order, so the exponent follows their sum and their difference drops out.

That is checked rather than assumed. On an interior generation of a straight-armed crown, the absolute difference between the daughters’ log lengths explains under two per cent of the exponent’s variation. The shape of a fork matters to the exponent only through how long its two branches are together.

The slope has a closed form #

The same linearisation, applied to the two-level form, gives the slope directly. Under equal stress it is $p^2(1-\lambda)/(6\ln 2)$, with $p$ the exponent the rule conserves at that mean ratio; under equal bending it is $p^2(4-\lambda)/\bigl(8(2+\lambda)\ln 2\bigr)$. At the planar crown those are 0.282 and 0.877, against 0.296 and 0.856 measured on crowns turned thirty degrees at every fork.

The local form’s slopes are $p^2/(6\ln 2)$ and $p^2/(4\ln 2)$, so the parent’s carrying its daughters’ lengths keeps exactly a fraction $1-\lambda$ of the local response under stress and $(4-\lambda)/(4+2\lambda)$ under bending. At the planar ratio that is 0.29 and 0.61. Those two fractions are the whole of the difference between the dashed lines and the solid ones in the earlier figure.

Across the whole family #

The stress slope stays between 0.250 and 0.299 from halving lengths to 0.9. The bending slope climbs from 0.457 to 1.571. The ratio between them is 1.63 at halving lengths, 2.90 at the planar crown and 6.29 at 0.9, so it grows exactly where the exponents of the two rules are closest and the old identification was weakest. The forms hold closely up to about 0.8 and fall away above it, where thirteen generations are too few for the arms of a slowly shortening crown to converge.

Under flow the slope is zero at every ratio, which returns the null answer from the start: a crown whose junction exponents do not follow its lengths at all was sized by something that does not read them.

How many junctions it takes #

A slope is an estimate, and on a real crown it is taken from however many junctions were measured. The question is how many it takes to name the rule at the planar crown, where the exponent names nothing, when every radius carries two per cent error.

Named by slope, against a dividing line taken from crowns measured exactly, a stress-sized crown is misnamed 28 times in a hundred from four junctions, 8.5 from sixteen, 2.3 from thirty-two and 1.5 from forty-eight. A bending-sized crown, whose slope is steeper and whose regression is tighter, is misnamed 1.5 times in a hundred from sixteen and never from thirty-two.

More junctions make the scatter worse #

Named by scatter from the same junctions, a stress-sized crown is misnamed 44 times in a hundred from four junctions and 88 from sixty-four. The curve runs the wrong way, and the reason is worth stating plainly: the error has made the stress junctions scatter like bending junctions, and more junctions only estimate that inflated scatter more precisely.

That is the general shape of a biased instrument, and it is the case a count carries no error makes about numbers reported without intervals, with a sharper edge. An estimator that is wrong converges on the wrong answer, and the sample size that would settle an unbiased one confirms the mistake instead.

Where the informative junctions are #

The slope depends on the spread of summed lengths across the junctions measured, so a sample of junctions whose daughters are all roughly the mean length says almost nothing, however many there are. A useful sample reaches both ends: forks whose daughters are both long and forks whose daughters are both short.

Choosing that sample is a decision about which junctions to admit, and the band decides the answer is the warning that such a decision moves what a fit returns. It is also the lesson which junctions say anything draws for the exponent itself, where the lopsided forks carry the information and the even ones do not. The informative junctions for a slope are a different set. They are the forks whose daughters agree with each other and disagree with the mean, which is not a shape anyone would pick out by eye.

What the measurement asks of a field study #

Three things per junction, rather than one. The parent’s radius and both daughters’ radii, which a study of the exponent already takes; both daughters’ lengths, which fitting the exponent does not need and which are rarely reported; and the parent’s own length, so that each daughter’s ratio is a ratio and not a length.

The last of those is the hard one, for the reason three rules one exponent gave: a branch is bounded by junctions that are not evenly spaced, and deciding where one generation ends is a convention. A slope is less sensitive to that convention than a mean ratio is, because a consistent misassignment shifts every summed log length together, but it is not immune to one that is inconsistent.

Where the thick daughters are #

One detail of the stiffness rule surfaced only at the edge of the range. At the widest spread measured, a half-width of 0.6, on a crown whose mean ratio is 0.9, ten junctions in 4,095 carry a daughter sized thicker than its parent, and such a junction conserves no exponent at all.

It happens only under bending and never under stress, because bending sizes on a length squared, and a daughter half as long again as a parent that is already nearly its own parent’s length is sized for a longer lever than the branch that holds it. Such junctions are counted rather than averaged, and they are rare enough not to move anything above. A crown sized for stiffness with real length variation would show them as its most obviously strange forks.

What this does not say #

It does not say that real branch lengths spread uniformly in log length, independently from fork to fork, or uncorrelated with the loads. It carries radius error as independent noise on each radius, which is the kindest case: a systematic error, such as the thickening a swelling at the fork puts on every radius read near a junction, would move all the exponents together and could tilt the slope rather than only loosen it.

It does not say that any tree is sized by stress or by stiffness. The null answer is the flow rule’s zero slope, and the two mechanical slopes are what a crown sized that way would give. A measured slope between them names neither rule, and a slope above the bending one names nothing on this list.

The claim, reduced #

A spread of branch lengths leaves each sizing rule’s mean exponent where one length ratio put it, scatters the junctions, and adds a second measurement that names the rule where the exponent cannot. At the planar crown a junction’s exponent follows its daughters’ summed log length with a slope of $p^2(1-\lambda)/(6\ln 2)$ under stress and $p^2(4-\lambda)/\bigl(8(2+\lambda)\ln 2\bigr)$ under bending, 0.30 against 0.86, zero under flow — and a two per cent error in every radius moves neither slope while it makes a stress crown scatter like a bending one.

What would refute it #

A flow-sized junction conserving anything but three. A regression slope that moves with the radius error as fast as the scatter does. A straight crown whose slope departs from the carried form by more than it departs from the local form. A mean length ratio at which the bending slope is not the steeper of the two. A difference between the daughters’ lengths that explains a real share of the exponent. Each is checked whenever the crowns are measured.

Still open: lengths that are not independent #

Every crown here draws each daughter’s length without regard to its parent’s or its sibling’s. A real tree may regulate its crown, so that a long branch carries short daughters and the lengths along a path compensate, and a compensating crown would feed a correlation into exactly the sum the slope is taken on.

The measurement is a crown whose daughters’ log lengths are drawn with a stated negative correlation to their parent’s, sized by each rule in turn: whether the slope under each rule moves, and whether the carried form still predicts it once the parent’s arm is allowed to know its daughters in advance. A slope that survives regulation is a measurement of the sizing rule. One that does not is a measurement of the regulation, and would have to be read as that.