A count carries no error

The duel between the two rules overlaps at twelve per cent of error on every radius, and a standard correction removes most of the displacement without moving that threshold. The overlap survived because it is a property of the question: fitting r₀ᵏ = Σrᵢᵏ at each junction, with a measured parent on one side and measured daughters on the other.

A tree’s radii contain more than their junctions. Every branch carries some number of tips, and that number is known exactly by counting. The question is what a tree’s radii say when they are read against it.

A radius is a power of its count #

Under either rule, with every tip the same size, the arithmetic is one line. A junction conserves rᵖ, so a branch’s rᵖ is the sum of its daughters’, and by induction up the tree it is the number of tips it carries. A branch carrying N tips has radius N^(1/p), exactly, with the tip’s radius as the unit.

So log r = (1/p)·log N. The exponent is the reciprocal of a slope — and the regressor in that slope is a count. A tree built at three has radii that rise as the cube root of their counts; a tree built at two, as the square root.

The noise sits on one side #

That is the whole advantage. A relative error on each radius adds scatter to log r and shifts its mean slightly, by the same amount on every branch, which is an intercept. The count has no error in it, so nothing is on the regressor’s side to drag the slope towards zero.

In a junction-by-junction fit, by contrast, the parent’s error and the daughters’ errors both sit inside the relation being fitted, which is what produces the downward displacement every earlier measurement of it has found. Changing the regressor changes where the noise is, not how much of it there is.

The tree #

The tree is grown by splitting a uniformly chosen tip fifty times, which gives 101 segments and 51 tips over eleven generations. Forty-five of its fifty junctions are even enough, by their tip counts, to have a daughter ratio of 0.6 or more under Murray’s law, and the median junction’s ratio is 0.91 — close to the informative band the duel was measured on.

Each replicate places the same relative Gaussian error the duel uses on every radius, three hundred replicates at each error, and the junction fit and the count fit are both run on the same measured radii. Any difference between them is the instrument.

One replicate at twelve per cent #

In the replicate drawn above, the tree built at three gives a slope of 0.3404, an exponent of 2.937, and the tree built at two a slope of 0.5071, an exponent of 1.972. On the same measured radii, the junction-by-junction fit reads 2.218 and 1.736.

The count is not perfect on one replicate — 2.937 is six hundredths short — and it does not need to be. Its errors are scatter about the truth rather than a displacement from it, which is the property an interval needs.

At every error measured #

At twelve per cent of error the count reads 2.996 and 1.997, with intervals of [2.81, 3.19] and [1.91, 2.08]. At twenty per cent, 2.998 and 1.997. At thirty, the highest error whose rows can be quoted, 3.015 and 2.001, with intervals of [2.52, 3.58] and [1.77, 2.24]. The count’s two intervals never overlap.

The junction fit on the same radii reads 2.060 and 1.661 at twelve per cent, overlaps from sixteen, and at twenty-five per cent reads 0.947 and 1.010 — the inversion the duel found, on a different tree.

Two margins at twelve per cent #

On this particular tree the junction fit has not quite overlapped at twelve per cent: its intervals, [1.81, 2.32] and [1.54, 1.79], clear each other by two hundredths. The duel’s designed band of fifty informative junctions overlapped at twelve; a Yule tree’s fifty junctions, a slightly different sample, overlap at sixteen.

The count’s intervals at twelve per cent clear each other by 0.73. At sixteen per cent, where the junction fit’s intervals overlap by a quarter of a unit, the count’s are still 0.63 apart. The comparison is not between two ways of just failing; it is between one instrument at its limit and one nowhere near it.

The interval, worked #

The count’s interval can be predicted before any replicate is drawn, which is a stronger statement than measuring it. A relative error of σ on a radius is a scatter of about σ in its logarithm. A regression slope through points with that scatter has a standard error of σ/√Sxx, where Sxx is the spread of the regressor — here the sum of squared deviations of log tip count over the 101 segments, 96.44.

At twelve per cent that is 0.12/9.82 = 0.0122 in the slope. The exponent is the slope’s reciprocal, so its error is the slope’s divided by the slope squared: 0.0122 × 9 = 0.110 for the tree at three, and a central ninety per cent of ±1.645 × 0.110 = ±0.18 about three, [2.82, 3.18]. The replicates give [2.81, 3.19]. For the tree at two the same arithmetic gives 0.0122 × 4 = 0.049 and [1.92, 2.08], against a measured [1.91, 2.08].

Where the quick arithmetic stops #

The same calculation at thirty per cent gives 0.30/9.82 × 9 = 0.275 for the tree at three and an interval of [2.55, 3.45]. The replicates give [2.52, 3.58]. The approximation that a relative error is a scatter of σ in the logarithm is a small-error one: the logarithm of 1 + σε is wider than σ and skewed once σ is large.

So the quick arithmetic understates the count’s real interval as the error grows, and the real interval is asymmetric, reaching 0.58 above three and 0.48 below. That matters for anyone quoting the count from a formula rather than from replicates: at large error the formula is the optimistic one, and the replicates are the answer.

What counting asks of a person #

A count is free of measurement error and not free of effort. Counting the tips a branch carries means following it outward to every shoot, on every branch whose radius is measured — easy on a winter crown photographed against the sky, hard in a dense canopy where tips hide behind one another.

That is a different cost from the junction fit’s, which asks for three radii measured at one place. It moves the care from the callipers to the counting, and a count that misses tips is a count with its own error, which the last section of this essay takes up.

Where the count’s information comes from #

That calculation says what the count needs, and it is not what a junction fit needs. The informative junctions are the ones whose daughters are comparable, because their leverage on the exponent is large. The count’s information is the spread of tip counts across the whole tree, and every segment contributes to it — the twigs at a count of one as much as the trunk at fifty-one.

So the junctions a junction fit has to discard, the lopsided ones and the twigs, are what holds the count’s regression in place. A tree offers its tips in quantity, and the instrument that reads them uses exactly what the other one throws away.

The count widens and does not move #

The count’s intervals do widen with the error — from a width of 0.38 at twelve per cent to 1.06 at thirty on the tree at three — because scatter in log r is scatter in the slope. What they do not do is move. The tree at three’s mean is 2.999 at one per cent, 2.996 at twelve, 2.998 at twenty and 3.015 at thirty.

The small rise at thirty per cent is not a displacement of the fit. At that error eight of the three hundred replicates had a radius pushed to zero or below and were refused, and a mean over the survivors is a mean over the replicates whose errors happened to be milder.

At a quarter of a radius #

At twenty-five per cent the scatter is large enough to see from across a room, and the count still reads 2.879 and 1.945 on the replicate drawn. The junction fit on the same radii reads 1.121 and 1.147 — the tree built at three reading lower, the two rules not only merged but swapped.

A person with photographs of fifty branches, measuring each to a quarter of its radius and counting its tips, has an instrument that separates the two rules. The same person fitting the junctions has one that reports the wrong rule.

Small trees and large ones #

The size of the tree matters, and in an unexpected direction at the top. On ten junctions the count keeps the trees apart to twelve per cent and overlaps at twenty; on twenty junctions, apart to twenty-five, overlapping at thirty. On fifty junctions or more it never overlaps at any error whose row can be quoted.

The junction fit overlaps at twelve per cent on ten and twenty junctions and at twenty per cent on fifty and two hundred. More junctions help it, as the window that closes found they do, up to a point; the count needs far fewer to do far better.

The ceiling is the error model #

A larger tree is quoted to a smaller error: thirty-five per cent on ten and twenty junctions, thirty on fifty, twenty-five on two hundred and a thousand. That is not the count failing. A relative Gaussian error of thirty per cent pushes a radius to zero or below in 8 of 300 replicates on a fifty-junction tree, 51 on two hundred and 160 on a thousand, simply because a larger tree has more radii for the tail of the error to reach.

A radius measured to thirty per cent is not a measurement anyone would publish, so the ceiling says more about the Gaussian model than about trees. Below it, on any tree of fifty junctions or more, the count’s two intervals are apart at every error.

The count’s own error #

A count is error-free only as a count of equal tips, and that is the assumption the instrument lives on. Give each tip a radius drawn log-normally with a spread of 0.2 and, with no measurement error at all, the count reads 2.832 on the tree at three and 1.948 on the tree at two; at a spread of 0.3, 2.654 and 1.888.

The junction fit on the same trees reads 3.000 and 2.000 exactly. Every junction still obeys its rule, because the rule is about junctions; only the proportion between a branch’s radius and its count has been broken, and that proportion is what the count reads.

The reverse of the junction fit’s weakness #

So the two instruments fail in opposite places. The junction fit is exact whatever the tips look like and is displaced by measurement error. The count is undisplaced by measurement error and is displaced by tips of unequal size, by an amount that grows with their spread.

That is a better situation than having one instrument. A tree read both ways, with the two readings compared, says which of the two errors dominates it — which neither reading says alone.

Both errors at once #

With twelve per cent of measurement error added to tips of spread 0.2, the count reads 2.828 [2.66, 3.02] on the tree at three and 1.946 on the tree at two — displaced by the tips exactly as much as with no error, and still well apart. At a spread of 0.3 it reads 2.651 and 1.886.

The junction fit on the same radii reads 1.995 [1.77, 2.26] and 1.642 [1.51, 1.78], overlapping by a hundredth. With both errors present the count is the one reading a tree at three as nearer three than two.

A protocol that treats tips differently #

A second way a count can be misled is a protocol that measures twigs one way and branches another — calipers on the branches, something finer on the tips. Put every non-tip segment five per cent fat and read the tree at three: the count with the tips in reads 2.838, the junction fit 2.693, and the count with the tips left out reads 3.000 exactly.

The repair is arithmetic. With the tips excluded, every remaining radius carries the same factor, and a common factor in a regression of log r on log N is an intercept. At ten per cent of offset the numbers are 2.700, 2.432 and 3.000; the tree at two gives 1.862, 1.738 and 2.000.

The growth record reads the same way #

A year of growth adds wood to every branch that gained tips, and the count reads that too. Under Da Vinci’s rule a branch’s area is proportional to its tip count, so the area added per new shoot is exactly one tip’s area in every branch, however thick: its slope against log radius is 0.000 in a season spread across the crown and 0.000 in one concentrated in a single leader.

Under Murray’s rule the area per shoot falls with radius, with slopes of −0.925 and −0.514. The ring alone, not divided by the shoots it fed, has a slope of 0.972 in one season and 0.000 in the other under Da Vinci’s rule — it moves with where the tree put its growth as much as with the rule, which is why the count has to be in the ratio.

Where the twelve per cent belonged #

The duel’s overlap was stated as the error at which the two rules become one measurement. It is the error at which they become one measurement of junctions. The same radii, read against a quantity measured without error, are one measurement of nothing until the error model itself breaks down.

That has been corrected where it was said. It changes the practical advice from which junctions to measure and how carefully to what to count beside them, and it moves the burden from the instrument to an assumption — equal tips — that a person can check on the tree.

A few lopsided junctions #

One more comparison is worth recording because it is stark. On a thousand-junction tree only two junctions have tip counts differing by more than a factor of thirty-seven, the junctions a band of twigs is made of. Read through their six segments at twelve per cent of error, the count gives 3.017 for the tree at three and 2.006 for the tree at two; the junction fit on the same junctions gives 1.333 and 1.049.

Six segments are not a sample to publish from, and the intervals say so. The point is only that the count’s advantage does not come from choosing good junctions — it holds on the junctions a junction fit cannot use at all.

What this does not establish #

That a real tree’s tips are the same size, which is exactly what the count’s failure prices and nothing here measures. The tree is a Yule tree, one draw of one random process; a real crown’s topology differs, and its tips include dead and broken shoots whose count is itself uncertain. The measurement error is the Gaussian, relative, independent error the duel uses, which makes a radius non-positive often enough past thirty per cent that those rows are reported with their refusals rather than quoted.

It also says nothing about why a tree would obey either rule. A static tree read by its count can say which exponent it has; it cannot say what sized it.

What would withdraw it #

A count fit that returns anything but the built exponent with no error. At twelve per cent on fifty junctions, the count’s intervals overlapping, or its mean displaced from its truth by more than a twentieth. A count with the tips left out that does not return the exponent exactly under an offset on every other branch. An area per new shoot that varies with radius under Da Vinci’s rule. Each is run every time the measurement runs.

Why a tree would have two #

The count establishes that a tree’s radii can tell the two exponents apart. It cannot say why a tree would obey either, and Da Vinci’s two has had no derivation here — only a name, and the phrase “the mechanical answer”. The next test is a count that is itself wrong — tips lost to pruning or breakage, which removes from N branches that still carry the wood of what they fed — and its test is how many lost tips the count can absorb before its advantage is gone. The question of mechanism belongs with the cube law, where a cube law with a lever arm derives an exponent of two from a stress rule and finds the length of a tree’s branches deciding which exponent it should have.