Three rules, one exponent

Three arguments now size the same symmetric crown, and each gives an exponent that its junctions conserve. Murray’s flow rule sizes for the least work in moving fluid and conserves three. Equal bending stress sizes for the same stress at every base and conserves 3/(1 + ℓ). Equal deflection sizes so that every branch bends by the same share of its length and conserves 4/(1 + 2ℓ). In each of these ℓ is log₂(1/λ), how many halvings of length a generation is worth.

A measured exponent is routinely offered as evidence for one of them, which raises a question that has an answer: given an exponent and whatever else a person could measure on a tree, can the rule that sized it be named?

Only one of the three reads the lengths #

The first thing the figure says is that the flow rule is flat. Murray’s derivation balances the work of driving fluid through a tube against the cost of keeping the tube’s contents alive, and neither term contains a length — the length cancels, which is why the cube law is a rule about radii alone and holds at every junction of a tree whatever its shape.

The two mechanical rules are not flat, because a lever arm is a length. That single difference is what makes the identification possible at all: two curves and a horizontal line can be told apart where all three are far enough from each other, and cannot be where any two meet.

An exponent alone names nothing #

Take a measured exponent of 2.1. Under the flow rule it is a tree that missed three, which the measurement error on any real sample easily allows. Under equal stress it is a crown whose branches shorten by 0.74 a generation. Under equal deflection it is a crown whose branches shorten by 0.7266. Three different claims about the same tree, and the exponent has no way to choose.

So the exponent must be read with a length ratio beside it, and the length ratio is the harder measurement of the two. It requires the lengths of branches at successive generations, which on a real crown means deciding what a generation is.

The three rules in one sentence each #

Murray’s rule minimises the sum of the work needed to drive fluid through a branch and the metabolic cost of the blood or sap inside it, and the minimum puts the cube of a parent’s radius equal to the sum of the cubes of its daughters. Equal stress puts the bending stress at every branch’s base at one value, so that no branch is nearer breaking than any other. Equal deflection puts every branch’s sag at one share of its own length, so that no branch is nearer being in the way than any other.

Each is an optimisation with a different thing being economised, and none of them is a mechanism. A tree does not compute a minimum; at best a tree is the survivor of a selection that acted like one, and an optimum too flat to reach is the account of what happens when the minimum is broad enough that the selection has nothing to push against.

Where the three separate #

At a crown whose branches halve in length, the three predictions are 1.3333, 1.5000 and 3. The nearest pair are 0.1667 apart and the furthest 1.6667, so an exponent good to a twentieth names the rule outright.

That is the comfortable case, and it is comfortable because 0.5 is well below every crossing. The uncomfortable cases are not spread through the family; they sit at three specific ratios, and each has a reason.

The first band: stress against stiffness #

The two mechanical rules cross at λ = 2^(−1/2), the crown that fills a plane, where both give exactly two. Near that ratio nothing separates them, because they are the same tree there. At a precision of five hundredths the band runs from λ = 0.670 to λ = 0.736 — ten of the ninety-six ratios sampled.

And the band is wide because the crossing is gentle #

The difference between the two curves goes through zero smoothly rather than steeply, so a tolerance of five hundredths buys a band of 0.066 in the length ratio. That is the widest of the three bands, and it sits exactly where the most-quoted answer in the subject lives: a tree reported as conserving two, with branches shortening at about the planar ratio, is consistent with both mechanical rules and cannot be evidence for either.

The second band: stiffness against flow #

The stiffness curve is the one that climbs fastest, and it passes three at λ = 0.8909. So a crown whose branches shorten by about a ninth a generation is predicted to conserve three under equal deflection — the same number Murray’s rule gives at every ratio — and the two are inseparable near there. At a precision of five hundredths the band runs from 0.888 to 0.896, two of the sampled ratios.

This one is narrow because the stiffness curve is steep at that point: it is passing three while the flow line stands still, so a small change in the length ratio moves the gap quickly. Narrow does not mean unimportant, though, because a crown that barely shortens is a real shape.

The third band: stress against flow #

The stress curve also reaches three, but only as λ approaches one, and it approaches logarithmically slowly. At λ = 0.95 it reads 2.7933 and at λ = 0.99 it reads 2.9571. So the last band is a single ratio at the very top of the range, and it is the ratio at which the whole self-similar picture stops meaning anything: a crown whose branches do not shorten has no scale to it.

That third band is therefore the least interesting of the three and the one most likely to be met in a bad measurement, because a crown whose generations were assigned wrongly can easily report a length ratio near one.

The site’s own drawn tree sits in the first band #

The crowns drawn here have branches 0.74 of their parent’s length, a ratio chosen to look like a tree rather than to make an argument. At that ratio equal stress predicts 2.0915 and equal deflection 2.1404 — 0.0489 apart, inside a tolerance of five hundredths and therefore inseparable.

That is worth stating because it is the ordinary case rather than a contrived one. A crown that looks like a tree sits near the planar ratio, and near the planar ratio the two mechanical rules are the same rule to within the precision anybody measures.

The precision decides how much of the family is readable #

Reducing the measurement’s precision widens all three bands at once, and the growth is faster than the precision. At 0.01, two of ninety-six ratios are unreadable; at 0.02, five; at 0.05, thirteen; at 0.1, twenty-seven; at 0.2, seventy-six; and at 0.3 there is nothing left to read.

Why it is faster than linear #

Two of the three pairs meet by crossing, where the gap grows linearly with distance from the crossing, and halving the tolerance should halve the band. The stress and stiffness curves cross at a shallow angle, so their band is wide at every tolerance; and once the tolerance is large enough that the stiffness band and the flow line overlap over a stretch rather than at a point, the two bands merge and the count jumps.

At a tolerance of three tenths every ratio in the range is ambiguous, which is not a subtle statement: an exponent known only to within 0.3 is consistent with all three rules everywhere, and a great many published exponents are known no better than that. A count carries no error is this collection’s standing complaint about numbers reported without intervals, and an exponent without one is the same complaint with more arithmetic behind it.

The bands at a precision of one hundredth #

Tightening the measurement to a hundredth collapses the picture almost entirely. Only one band survives, stress against stiffness from λ = 0.707 to 0.714, and it is a single sampled ratio wide: the crossing itself, which no precision removes because the two curves are genuinely equal there. The stiffness-against-flow band disappears below a tolerance of about two hundredths, and the stress-against-flow band with it.

So the identification is not hopeless; it is expensive. An exponent known to a hundredth and a length ratio known well enough to place it on the curve would name the rule everywhere except at the crossing, and the crossing is one point rather than a region.

What the measured tree actually gives #

Fitting the exponent is the account of how an exponent gets estimated from a set of junctions, and it is not encouraging: the estimator is biased, the bias depends on which junctions were sampled, and the informative junctions are the lopsided ones — a junction whose daughters are nearly equal says almost nothing about the exponent whatever it is. The exponent an error moves puts numbers on how a stated error in each radius propagates, and a swelling at the fork on what the thickening around a junction does to a radius measured too close to it.

Two further biases push the same way, and both come from the previous essay. Every mechanical exponent is approached from below with depth, and a crown of seven generations reads two per cent under its planar limit and four per cent under its volume-filling one; and the accessible junctions on a real tree are the outer ones, which read furthest under. Two biases in the same direction are worse than two in opposite directions, because they add rather than partly cancelling.

The length ratio is the measurement nobody reports #

An exponent is reported constantly and a length ratio almost never, which is the practical obstacle rather than any of the arithmetic above. Measuring λ means pairing each branch with its parent and taking a ratio of lengths, and on a real crown a “branch” is bounded by junctions that are not evenly spaced: a limb may run three metres before forking and its daughter fork after thirty centimetres, and calling those one generation apart is a decision rather than an observation.

Two decisions make it worse. If short internodes are absorbed into their parents, generations become longer and λ is over-estimated, pushing the reading towards the ambiguous band near one. If every side shoot counts as a generation, λ is under-estimated and the crown is placed far below the planar ratio where everything separates cleanly — a comfortable answer arrived at by a convention. Nothing here says which is right; it says that the answer moves with the choice, which is the thing to report alongside it.

Which makes the flat rule the easiest to reject and the hardest to confirm #

There is an asymmetry worth naming. A measured exponent well away from three rejects the flow rule outright, whatever the length ratio, because the flow rule makes the same prediction everywhere. But an exponent near three confirms nothing, because two of the three curves pass through three somewhere.

So the usual shape of the evidence — a tree measured at 2.9, called consistent with Murray’s law — is the weak direction of an asymmetric test. The strong direction is a tree measured at 1.4, which no length ratio reconciles with the flow rule and which both mechanical rules reach at a crown whose branches shorten fast.

What a study that could settle it would need #

Three measurements rather than one. The exponent, to better than a twentieth; the length ratio, well enough to say which side of 0.7071 the crown is on; and the depth, so that the finite-crown shortfall can be taken off rather than absorbed into the answer. On a crown with fewer than about nine generations the third of those is the largest of the three corrections.

And a fourth, which is not a number: an argument that the crown is self-similar at all. Every exponent here belongs to a tree with one length ratio, and a crown whose lower branches shorten differently from its upper ones has no single λ to measure — which makes it not a crown this family describes rather than a crown with an intermediate exponent. The band decides the answer is the version of that trouble already measured on the estimator side, where the range of junction sizes admitted to a fit moves the exponent it returns.

What a spread of tolerance does to the answer, drawn #

A precision of a tenth is what a careful field study might reach, and at that precision more than a quarter of the family is unreadable. The band around the planar crossing has grown to run from 0.620 to 0.765, which covers the length ratios of most trees anybody would draw. That is the practical summary of the whole essay: at the precision available, the two mechanical rules are one rule over the part of the family that contains real crowns, and the useful distinction is between them together and the flow rule.

What the three rules share #

They agree about one thing, and it is the thing the flow rule is usually praised for. All three conserve some power of the radius at every junction, which is why a branching tree can be described by one number at all. That is a property of self-similar sizing under a self-similar load, not of any particular criterion, and it means the existence of a conserved exponent is evidence for far less than it is usually taken to be.

What the exponent’s value is evidence for is what this essay measures, and the honest answer is: for the rule, only in the company of a length ratio, and then only outside three bands covering about an eighth of the family.

What would have to be added to separate them #

The three rules differ in more than their exponents, and the other differences are where a decisive study would look. Equal stress and equal deflection predict different absolute radii for a given trunk, not merely different ratios, so a crown measured in millimetres rather than in ratios carries information the exponent throws away. The two also respond differently to the load moving from the tips onto the wood: under equal stress a planar crown carrying only its own weight conserves one, and under equal deflection four thirds, which is a far wider gap than either rule’s tip-load reading offers.

That suggests the readable measurement is not a mature crown at all but a comparison — the same species grown where wind dominates against where self-loading does, or the lower and upper thirds of one crown, where the share of the load carried by the wood differs. A difference between two readings is not subject to the depth bias in the same way, because both readings are taken at similar depths.

What is claimed, in one line #

A branching exponent names a sizing rule only with a length ratio beside it, and at thirteen of ninety-six length ratios sampled it does not name one even then — in three bands, where the two mechanical rules cross, where the stiffness curve passes three, and where the branches stop shortening.

What would withdraw it #

A length ratio at which the three rules give exponents the measurement claims to separate but does not. A band of ambiguity between stress and stiffness away from the crossing at 2^(−1/2). A pair of rules inseparable at a tolerance of 0.01 over more ratios than at 0.02. A crown whose junctions conserve no single exponent under one of the three rules. Each is checked every time the measurement runs.

Still open: whether a real crown has one length ratio #

Every statement here is about a crown with a single λ, and the identification turns on measuring it. Nothing above has asked what a crown with a distribution of length ratios does — whether its junctions still conserve a single exponent, and if so whether that exponent is the one its mean ratio predicts or something the spread pulls away from. The measurement is a crown whose every fork draws its two daughter lengths from a distribution of stated width, sized by each rule in turn: whether the junction exponents still fall in one generation-wide band, and how the trunk’s reading moves with the spread. A rule that survives a spread and one that does not are different claims about trees.