The band decides the answer

The previous phase found that a sample of fifty lopsided junctions, taken from a tree built at an exponent of exactly three, returns 1.69 with an interval of [1.56, 1.79] — excluding three, excluding two, and only a fifth wider than a good sample’s. The cause is a selection nobody applies on purpose: at a daughter ratio of a twentieth the parent is genuinely thicker than its larger daughter by 0.11%, against a measurement noise of 2.8%, so about half the measured junctions come out with the parent thinner than a branch it carries. Those have no exponent at all, every estimator drops them silently, and the half that survives is the half where noise inflated the parent.

The obvious next worry follows immediately and it is the one this essay tests.

A real tree is not a band of daughter ratios. It is all of them at once, and the lopsided ones vastly outnumber the even ones — a tree has one trunk fork and hundreds of twigs. So a single fit over a whole tree ought to be dragged towards the twigs’ 1.7, and the amount it is dragged ought to depend on how many twigs happened to be in the sample.

It is not dragged #

Two hundred junctions spanning daughter ratios from 0.05 to 1, from trees built at an exponent of exactly three with the standard two per cent measurement error, fit 2.85 to 2.99 across five seeds. The twig band alone, from the same trees with the same fitter, gives 1.60 to 1.73.

The prediction is refuted, and the refutation is more useful than the prediction would have been, because the reason is a property of least squares rather than a happy accident.

The leverage does it #

The residual a junction contributes to the fit moves with the exponent at a rate set by how far apart the candidate exponents’ predictions are at that junction — which is exactly the quantity the previous essay measured as the separation, and which collapses as the cube of the daughter ratio. At an even fork the two rules predict parent radii 0.154 apart; at a daughter ratio of a twentieth, 0.0012 apart.

A least-squares fit weights each observation by how much moving the parameter changes its residual, which is the square of that separation. So the weights across a tree are in the ratio 0.154² to 0.0012², which is more than sixteen thousand to one.

Computed band by band, the shares are:

• daughter ratio 0.9–1.0 — 41.8% of the tree’s leverage
• 0.7–0.9 — 32.7%
• 0.5–0.7 — 18.3%
• 0.3–0.5 — 6.2%
• 0.15–0.3 — 1.0%
• 0.05–0.15 — 0.06%

Three quarters of a whole tree’s leverage sits in the junctions with a daughter ratio above 0.7, and the twigs carry six hundredths of one per cent of it.

So the estimator is already doing the right thing, without anybody choosing it. Adding a hundred twigs to a sample of ten even forks changes the answer by essentially nothing, because the twigs contribute essentially nothing.

Which relocates the danger #

The result does not make the previous phase’s finding harmless. It relocates it, and the new location is worse.

The bias appears when a sample is restricted to the uninformative band, and the question is when that happens in practice. It happens whenever the sample is selected by something correlated with the daughter ratio — and the most obvious such thing is whether a person can reach the junction and put callipers on it.

On a standing tree, twigs are at eye level and in the hand. The trunk fork is twenty metres up. A sample gathered by walking through a wood with callipers is not a mixed sample; it is a twig sample, and a twig sample returns 1.7 with a tight interval from data generated at 3.

So the practical advice inverts. It is not “take a big sample and let the mixture sort itself out” — that works, but it is not what anybody does. It is “measure the few junctions that are worth measuring”, which on this arithmetic means the symmetric ones, and one of them settles the question at the top of the range.

The retention rate, which is the diagnostic #

The previous phase ended with a recommendation: report attempted and retained, not retained alone. This essay measures what that number does across the range, and it tracks the bias exactly.

• daughter ratio 0.9–1.0 — median exponent 3.00 — 100% of junctions have one
• 0.7–0.9 — 2.98 — 100%
• 0.5–0.7 — 2.95 — 99%
• 0.3–0.5 — 2.73 — 78%
• 0.15–0.3 — 2.11 — 56%
• 0.05–0.15 — 1.58 — 53%

Two columns of the same fact. Where the parent’s genuine margin over its larger daughter exceeds the measurement noise, every junction has an exponent and the median is the true one. Where it does not, half the junctions are impossible, the survivors are the ones noise inflated, and the median collapses.

The value of the retention column is that it needs no model. A person reporting a branching exponent knows how many junctions they measured and how many the fitter could use; the ratio of those two is one integer over another and it says whether the answer is a measurement or a selection. A sample that retained 53% of its junctions is not a sample with a slightly larger error bar. It is a sample whose answer is wrong by more than the distance between the hypotheses being tested.

Why the median and not the fit, band by band #

The table above reports a median of per-junction exponents rather than a fitted exponent per band, and the choice is not cosmetic.

A per-junction exponent is the exact solution of r₀^k = r₁^k + r₂^k for that junction, which exists whenever the parent is thicker than its larger daughter and does not otherwise. So the median of the per-junction exponents in a band is a statement about that band’s junctions individually, and the count of junctions that have one is available as a by-product — which is the retention column.

A fitted exponent per band would hide both. It returns a number whether or not the junctions in the band are individually solvable, because the fit minimises a residual rather than solving an equation, and a residual can be minimised over data containing impossible junctions without any indication that they were there.

That is the same distinction the site keeps running into in other subjects: an instrument that always returns a value is not the same as an instrument that measures. The angle recovery refuses when no divergence makes a pair the closest. The comb readout refuses when no spacing wins by a band. The per-junction exponent refuses when a branch is thinner than what it carries — and the count of refusals is the diagnostic.

What a mixed sample does to the interval #

The point estimate survives a mixed sample. The interval does not, and it moves in the direction that matters.

Bootstrapping the whole-tree fit gives an interval of [2.80, 2.91] on two hundred junctions. The even-fork band alone, on the same number, gives [2.88, 2.99]. So adding a hundred and fifty uninformative junctions to fifty informative ones does not widen the interval much — it barely narrows it either, which is the leverage argument again — but it does shift it, by about a twentieth, downwards.

That shift is small and it is not nothing. It is the twigs contributing their six hundredths of a per cent, biased, and it says the correct summary is not “a mixed sample is safe” but “a mixed sample is safe in proportion to how much of it is informative”. A tree sampled with ten even forks and a thousand twigs would be dragged more than one sampled with fifty and a hundred and fifty, because the weight is a total rather than a share.

The version of the advice that survives all of this is therefore about a ratio rather than about a count: what matters is the leverage of the informative junctions in the sample against the leverage of the biased ones, and both are computable from the daughter ratios alone before any exponent is fitted. A person with a list of measured junctions can work out, in advance, whether their sample can answer the question — which is the same service the internode counts and specimen counts elsewhere in this collection provide, in a subject where nobody usually asks.

The limit of the claim #

All of this is arithmetic about a model. It says what a junction could report if the model is right, and what a sample of junctions does to an estimator under a stated measurement error. It does not say Murray’s derivation applies to a tree, and this site’s own essay on fitting the exponent is careful that a fitted value departing from three is the interesting case rather than an error.

Two specific assumptions carry the numbers. The measurement error is two per cent of a radius, stated rather than measured, and every count in the thread scales as its square — a reader who can do one per cent should divide the sample sizes by four and will find the retention rates rise correspondingly. And the daughter ratios are drawn evenly across the band, which no real tree does; a real distribution concentrated at the lopsided end has less leverage in total than the one measured here, though its shares by band are unchanged.

What does not depend on either is the shape: leverage falls as the sixth power of the daughter ratio, retention falls with it, and a sample selected by convenience is selected against information. That is a statement about a class of measurement rather than about trees, and it is why this essay sits in a collection about phyllotaxis at all.

Why an even fork is rare, and what that costs #

The arithmetic says one even fork settles the question. It is worth asking how often a tree provides one, because “one is enough” is only useful advice if one is available.

A branching architecture that divides its flow evenly at every junction is a particular kind of tree and most are not. The common case is a trunk that sheds branches — a dominant axis with subordinate ones coming off it — which puts every junction at a small daughter ratio by construction. Species with genuinely dichotomous branching, where a shoot divides into two comparable daughters, are the ones this measurement is cheap on, and they are a minority.

So the practical position is uncomfortable in a specific way. On a dichotomous tree the exponent is settled by a handful of junctions and the measurement is almost free. On an excurrent tree, where the interesting architecture is, the informative junctions are the few near the base and they are the ones a person cannot reach — and the thousands that are reachable carry, between all of them, less weight than one fork at the crown.

That is not a limitation of the estimator and no better estimator removes it. It is the geometry of the question: every exponent predicts nearly the same parent radius at a lopsided junction, so a lopsided junction contains almost no information about the exponent, however carefully it is measured and however many of them there are. The right response is to say so in the sample design rather than to hope that a large n compensates, because it does not — it is the one case where a bigger sample buys almost exactly nothing.

What it adds to the branching thread #

The thread now has three results and they fit together.

Which junctions could ever say anything: the separation between the two rules collapses as the cube of the daughter ratio, so one even fork settles it and two thousand twigs do not.

What happens when the ones that cannot are measured: a confident wrong answer, because half of them are impossible and the survivors are biased.

And what happens when everything is measured: nothing bad, because least squares is already weighting by the first result — which means the danger is never the mixture and always the selection.

The third is the one that changes what a person should do. The first two invite the reading “be careful with twigs”; together with the third the advice is sharper. Take whatever junctions are available and report the retention rate — or take the few good ones deliberately. What must not be done is to take the ones that are easy to reach and treat the number that comes out as though it came from a tree.

The number to report #

If one line comes out of this thread it is: report attempted and retained.

It costs nothing, it needs no model, it is two integers, and it separates a measurement from a selection in a case where nothing else in the output does. A sample that retained every junction has an answer. A sample that retained half has a number produced by which half survived.

That the same diagnostic works on this site’s sequence instrument is a coincidence of shape rather than of subject, and it is the reason the two threads ended in the same phase. A readout with a refusal in it produces two numbers — how many were attempted and how many returned — and the second on its own is never enough.