Which junctions say anything

There are two candidate rules for how a branching network divides. Murray’s law says the cube of the parent radius equals the sum of the cubes of the daughters, and it comes from minimising the work of moving fluid against the cost of maintaining the tube. Da Vinci’s rule says the same with squares, and it comes from conservation of cross-section — the observation, five centuries old, that a tree’s branches at any height together have about the trunk’s area.

An earlier essay in this collection fits the exponent rather than assuming it, which is the right method. This one asks a question that comes before the fit: which junctions in a tree could distinguish the two rules at all?

The answer is that almost none of them can, and which ones can is decided by the shape of the junction rather than by anything the measurer does.

The arithmetic #

At a junction with daughters of radius $r_1$ and $r_2 = \gamma r_1$, an exponent $p$ predicts a parent radius

$$\frac{r_0}{r_1} = \left(1 + \gamma^{\,p}\right)^{1/p}.$$

At an even fork — $\gamma = 1$, two equal daughters — the two rules predict $\sqrt{2} = 1.4142$ and $2^{1/3} = 1.2599$. They differ by 0.154, which on a two-per-cent measurement is about eight standard errors. One junction settles it.

As the junction becomes lopsided the two predictions converge, and they converge fast, because both tend to $r_0 = r_1$ as the small daughter vanishes. Every exponent predicts the same trunk when what comes off it is a twig.

A factor of two thousand across the range, and nothing in it is measurement error or model choice. It is the geometry of the question.

How fast they converge, exactly #

The table gives the collapse and the rate behind it is worth writing down, because it is the quantity that tells a person what a different junction would cost without measuring one.

Expand the prediction for a small daughter. When $\gamma$ is small, $\gamma^{\,p}$ is smaller still and the parent radius is $1 + \gamma^{\,p}/p$ to first order. So the two rules differ by $\gamma^2/2 - \gamma^3/3$, and for a lopsided junction the second term is negligible: the separation goes as the square of the daughter ratio, halved.

Check it against the table’s own ends. At a daughter ratio of a twentieth the formula gives 0.00121 against a measured 0.0012, and doubling the ratio to a tenth multiplies the separation by 3.85 rather than by 4, which is the cubic correction beginning to show.

That fixes the exchange rate. The number of junctions needed goes as the inverse square of the separation, so it goes as the inverse fourth power of the daughter ratio: halving the ratio of a lopsided junction costs sixteen times as many of them. From a fifteenth to a twentieth is a factor of three; from a fifth to a twentieth is a factor of two hundred and fifty-six.

Two things follow that the table shows and does not say.

The collapse is quadratic rather than exponential, which is why the requirement is merely two thousand rather than unmeasurable — a twig band is uninformative in practice and not in principle. And the fourth power is why the range is so wide: nothing about the geometry is extreme, and a fourth power over a factor of twenty in ratio is a factor of a hundred and sixty thousand in leverage all by itself.

It also says where the useful boundary is. Setting the requirement at ten junctions — a morning’s work rather than a career’s — puts the ratio at about a half, so the practical form of the rule is measure the forks whose daughters are within a factor of two of each other, and the arithmetic behind that sentence is one expansion.

The counts, and how they are got #

The sample size follows the same arithmetic every “how many would it take” figure in this collection uses, so it is worth writing out once here.

A junction gives one observable ratio, $r_0/r_1$, and it carries the error of two measurements — so its standard error is $\sqrt{2}\,\sigma$ for a relative error $\sigma$ on a radius. Averaged over $n$ junctions that is $\sqrt{2}\,\sigma/\sqrt{n}$. Requiring the gap between the two predictions to be two of those gives

$$n > \frac{8\sigma^2}{\Delta^2}.$$

Every count above assumes $\sigma = 2\%$, which is a stated assumption rather than a measurement of anybody’s calipers. It scales as the square, so an instrument twice as good quarters every count in the table — and one twice as bad quadruples them.

One side of this comparison is a hypothesis rather than a second measurement, and that is why the formula is not the two-sample one this site uses for the divergence sequence. Writing the two-sample form here first put every count out by a factor of four, in the safe direction, which is exactly the sort of error a table of plausible-looking integers hides.

Why the informative junction is the rare one #

The uncomfortable half is what a real tree offers.

A tree’s junctions are mostly not even forks. The common case is a trunk or a major limb shedding a smaller branch — a large $r_1$ and a small $r_2$ — and the even bifurcation is the exception, concentrated at the tips where the radii are hardest to measure and where the whole structure is least like a transport network.

So a sample of junctions taken as they come is a sample dominated by the ones that say almost nothing. Fifty junctions collected without regard to shape can carry less information about the exponent than five chosen ones, and the ratio between those two outcomes is not a subtlety — at the extremes it is three orders of magnitude.

That is a statement about which junctions to measure, and it is actionable in a way “measure more of them” is not.

The fit is not the problem #

It would be easy to read this as a criticism of fitting, so the control is worth stating: the fitter works.

Built at an exponent of 2.2, fifty even forks return 2.22. Built at 2.6, they return 2.62. Built at 3.4, 3.41. The machinery recovers whatever it is given and does not have three baked into it, which is the control the site’s own fitting essay already insists on.

What this essay adds is upstream of the fit. A fit can only extract information the data contains, and the table above is a statement about how much that is. A perfect fitter on a sample of twigs is a perfect fitter on nothing.

What it would take to settle the two rules #

Put the numbers together and the experiment is small.

Five even forks, measured to two per cent, distinguish Murray’s law from Da Vinci’s rule with room to spare. Twenty gives a fitted exponent with an interval of about ±0.1, which is enough to say whether a network sits at 2, at 3, or at neither.

That is a morning’s work on one tree, and it is a much smaller undertaking than the literature’s habit of measuring every junction in a specimen suggests. The reason it is smaller is not a better method; it is choosing which junctions to measure, which costs nothing and is decided before the calipers come out.

The catch, and it is the subject of the next essay, is what happens if the choice is not made — because a sample of uninformative junctions does not return a wide answer with an honest interval. It returns something worse.

Why both rules agree at a twig, in words #

The collapse of the difference has a physical reading, and it is worth having because it says the effect is not an artefact of writing the rule as a power.

Both rules are conservation statements. Murray’s conserves the sum of $r^3$, which is proportional to the flow a Poiseuille tube can carry for a given pressure gradient; Da Vinci’s conserves the sum of $r^2$, the cross-sectional area. When one daughter is tiny, it contributes almost nothing to either sum — a twentieth of the radius is a four-hundredth of the area and an eight-thousandth of the cube — so both rules reduce to the same statement: the parent is the same size as its large daughter.

That is a true statement about a tree and it is a statement both hypotheses make. A junction where a hair comes off a trunk says only that the trunk is a trunk.

The even fork is the opposite case: both daughters contribute fully, the two sums are being taken over comparable quantities, and the difference between summing squares and summing cubes is at its largest. It is the only place the two rules are being asked different questions.

So the information is not spread thinly across a tree — it is concentrated at the bifurcations where the two branches are comparable, and those are a specific, identifiable, minority population.

What the model does and does not assume #

Two limitations, because the arithmetic above is about a model rather than about a tree.

It assumes exactly two daughters. Real junctions are sometimes three-way, and the same analysis extends — the observable is $r_0$ against the several $r_i$ — but the information content changes, and this collection has not computed it. A three-way junction with one large and two small daughters is, for these purposes, close to a lopsided two-way one.

It assumes the radii are the quantities the rule is about. Murray’s derivation is about the tubes carrying the flow, and in a tree the conducting tissue is an annulus inside the branch rather than the whole cross-section. If the ratio of conducting area to total area varies systematically with branch size, the measured exponent is a mixture of the transport rule and that variation — which is a real confound, is well known, and is not addressed by measuring better junctions.

So the honest scope: this is arithmetic about what a junction of a given shape could tell if the model applies. It says nothing about whether the model applies, and the site’s existing essays on Murray’s law are careful that a fitted exponent departing from three is the interesting case rather than an error.

The cost curve, and where it turns #

The table is a set of points on a curve, and the shape of the curve is what a person planning a measurement needs.

The difference between the two predictions falls roughly as $\gamma^{\,2}$ for small $\gamma$ — both rules expand $(1+\gamma^p)^{1/p} \approx 1 + \gamma^p/p$, so the gap is $\gamma^2/2 - \gamma^3/3$, dominated by the square. The sample size, going as the inverse square of the gap, therefore grows as $\gamma^{-4}$.

A fourth power is steep enough that the useful range is short. Halving the daughter ratio multiplies the required sample by sixteen. Between $\gamma = 1$ and $\gamma = 0.4$ the cost goes from one junction to two; between 0.4 and 0.1 it goes from two to a hundred and forty-eight.

So the practical rule is a threshold rather than a gradient: junctions with $\gamma$ above about 0.4 are cheap and junctions below about 0.2 are hopeless, and there is not much in between. A field protocol can be stated in one line — measure the forks where the two branches look comparable, skip the rest — and that line captures nearly all of the available information.

What a survey should record #

The consequence for a study is a single extra column, and it is one that is almost never reported.

Record the daughter ratio at every junction measured. Not because it is interesting in itself, but because it converts a count of junctions into an amount of information — and without it, a paper reporting “148 junctions measured” cannot be distinguished from one reporting the equivalent of two.

With the column, a reader can compute the effective sample size directly: sum $1/\Delta(\gamma_i)^2$ over the junctions and compare it with the threshold. Without it, the sample size is a number of measurements rather than a measure of evidence.

That is a cheap recommendation and it has a precedent in this collection. The survey specification for spiral counts asks for the counting radius to be reported for the same reason: a count is worth $221°/mn$, so a count of 34 and 55 at a large radius is worth far more than one of 2 and 3 near the middle, and a study that reports counts without radii has thrown away the weights.

The general form #

The pattern is worth extracting, because it applies well beyond trees.

Two hypotheses differ by a quantity that depends on where in the sample space the observation is taken — which is the same shape as a lag window that has to reach a comb’s third tooth. The sample size needed goes as the inverse square of that difference. So the design question — which observations to take — enters the cost as a square, while the effort question — how many to take — enters it linearly.

That asymmetry means choosing what to measure is almost always worth more than measuring more, and by a large factor. Here the factor is two thousand between the best observation and the worst, and it is available for free to anyone who works out the difference before collecting.

This collection has made the same point twice in different subjects now. A parastichy count is worth about $221°/mn$ in pinning a divergence angle, so a count of 34 and 55 is worth vastly more than a count of 2 and 3 — same instrument, same effort, different observation. And a divergence sequence read on a stem in the wrong rate regime carries nothing, however many internodes it has.

Ask what an observation could settle before taking it, and the answer is usually an integer that decides whether the study is a morning or a year.

What this does not settle #

It is worth ending on what remains open, because the essay’s conclusion is comfortable and comfortable conclusions need their limits stated.

The calculation says a handful of even forks would distinguish an exponent of 2 from an exponent of 3 if the network obeys a single exponent. It says nothing about what happens when it does not, and this collection has already found that organs frequently do not: a fir cone and a capitulum have no single falloff exponent, a fitted one turns out to be a harmonic mean of what the individual steps report, and two measurements cannot show a varying exponent at all.

A tree is at least as likely to be in that position. If the exponent varies with branch order — plausible, since the conducting fraction and the mechanical demands both change from trunk to twig — then five even forks return a well-determined number that is an average of several, and the interval around it will not say so.

The remedy is the same one the exponent thread arrived at elsewhere: measure the even forks at several branch orders separately and compare, rather than pooling them. That costs a few more junctions and it is the difference between reporting a tree’s exponent and reporting a number.

And the arithmetic here is about two specific hypotheses. A study asking the open question — what exponent does this network have — is not choosing between 2 and 3, and its sample size is set by the precision it wants rather than by a gap between two values. That calculation is different and it is kinder: the standard error on a fitted exponent falls as $1/\sqrt{n}$ in the ordinary way, and even lopsided junctions contribute something to it.

What does not change is the ranking. The even forks are still where the exponent is best determined, by a factor that is large at every asymmetry, so the advice to choose junctions rather than to accumulate them survives the change of question. It is only the two-thousand-fold figure that belongs specifically to the two-hypothesis case.

What a whole tree does #

The obvious worry after this essay is that a real tree mixes informative and uninformative junctions, so a single fit ought to be dragged towards the twigs’ 1.7. Measured, it is not: two hundred junctions spanning the whole range of asymmetry fit 2.85 to 2.99 against a true 3.

So the danger is not a mixed sample. It is a sample selected by what a person can reach, and a sample selected that way is not merely uninformative — it returns a confident wrong number, which is the finding that follows this one.