Phyllotaxis
the spirals are counted, not admired
Field

Branching and transport

A network built to move fluid for the least work obeys a cube law at every junction. That is checkable on a real tree, though not from the angles: the same cost fixes those too, and the rival rule turns out to predict every one of them.
A branching tree in which every junction obeys the cube law. r₀³ = r₁³ + r₂³ at all 63 junctions, to 2e-16. The widths in the drawing are the radii the law gives, not widths chosen to look right.

The cube law

A branching network built to move fluid for the least work obeys one relation at every junction — the cube of the parent radius equals the sum of the cubes of the daughters. It is a minimisation result, it is checkable on a real tree, and it is the rare biological rule with a derivation.

 The exponent fitted from the junctions, rather than assumed. Sweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed.

Fitting the exponent

Assuming the exponent is three and reporting the error says how far the data is from that assumption. Fitting the exponent and reporting what it comes out as says what the network is doing — and an estimator has to be shown returning something other than three, or it is not a fit.

 An L-system after 4 rewrites of two rules. X → F[+X]F[-X]+X and F → FF, walked by a turtle turning 22.5°. 130 segments, and not one of them knows anything about light, water or auxin.

L-systems describe, they do not explain

Two rewriting rules and a turtle produce something indistinguishable from a plant, and there is no plant in it — no light, no water, no auxin, no mechanics. That is worth demonstrating precisely because the output is so convincing.

 The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

Which junctions say anything

Da Vinci's rule and Murray's law differ by 12% at an even fork and by a tenth of a per cent at a twig. So one even fork settles which is right, and two thousand twigs do not — a factor of two thousand across a tree, decided entirely by the shape of the junction and not by how carefully it is measured.

 The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

A sample that is confidently wrong

Fifty lopsided junctions from a tree built at an exponent of exactly 3 return 1.7, with an interval that excludes 3 and excludes 2 as well. The sample carrying almost no information does not give a wide answer — it gives a narrow wrong one, and the cause is a selection nobody applies on purpose.

 A tree built at an exponent of 3, measured band by band. Six bands of daughter ratio, 300 junctions in each, all from trees built at an exponent of exactly 3 with the same 2% measurement error. The median implied exponent falls from 3.00 at an even fork to 1.58 at a twig, and the share of junctions that have any exponent at all falls from 100% to 53% over the same range. The rule across the top is a single fit over 200 junctions spanning the whole range: 2.85. A mixed sample is safe because least squares already weights by leverage — 42% of it sits in the most symmetric band and 0.06% in the twigs. The sample that is not safe is the one a person can reach.

The band decides the answer

A fit over a whole tree's junctions returns the exponent the tree was built at, even though most of its junctions are from bands that on their own return 1.6. Least squares is already weighting by leverage. The dangerous sample is not the mixed one — it is the one a person can reach.

 A tree built at 3, measured to 2%, reads 2.957 on the informative band. The exponent recovered from 100 junctions of a tree built at exactly 3, against the relative error placed independently on the parent and on both daughters — half a per cent is a machined section under a microscope, one to two per cent is callipers on a clean branch, five is a branch that is not round, ten is a radius read off a photograph. Each line is a band of daughter ratio and the whiskers are the central 90% of 300 replicate samples. Every band is displaced downward at every error and never upward: at 2% the informative band read 2.957 and the informative band read 2.957. Below, the same rows with the displacement and the spread drawn as separate bars, because only one of them falls when more junctions are measured.

The exponent an error moves

Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone.

 One junction's bias moves 1.36-fold across the range its leverage moves 308,352-fold, at an exponent of 3. The bias a single junction of daughter ratio γ contributes to a least-squares fit, as a multiple of the squared measurement error, drawn against the leverage that junction carries — both computed from the expansion about a true exponent of 3 rather than fitted to anything. Across the whole range from an even fork to a twentieth the bias moves by a factor of 1.36 and the leverage by a factor of 308,352, and the uninformative junction contributes the larger share: 6.00σ² at γ = 0.05 against 4.50σ² at an even fork.

The fragile junctions are the informative ones

That is the obvious worry once the radii are uncertain, and it is false. Across the whole range of asymmetry a junction's contribution to the bias moves by a factor of 1.36 while its leverage moves by a factor of 308,352, so the junction that says nothing damages the answer as badly as the one that says everything — and a sample is spoiled by counting rather than by weight.

 Between 3% and 5% of radius error, no sample size answers — 50 junctions among them. One row per error level. The pale bar is the sample sizes whose interval is narrow enough to state a claim from — half-width under ±0.25 and excluding 2 — and it starts where precision arrives. The second bar is the sample sizes whose interval still contains the 3 the tree was built at, and it ends where the displacement overtakes the width. Where the two overlap there is a usable window; at 5%, 7%, 10% they do not overlap at all, so below 50 junctions the answer is too wide to state and above 30 it no longer contains the truth.

The window that closes

The spread of a fitted branching exponent falls as the reciprocal root of the sample and its displacement does not fall at all, so there is a count past which every further junction buys confidence and no accuracy. Between three and five per cent of radius error the count arrives before the answer does, and no sample size both states a claim and contains the truth.

 Murray's law and Da Vinci's become one measurement at 12% of radius error on the informative band. Two trees measured the same way: 50 junctions of daughter ratio 0.6–1, 300 replicate samples at each of 15 error levels, one tree built at exactly 3 and one at exactly 2. Each shaded band is the central 90% of the recovered exponents. Both run downward, but the tree at 3 runs down faster — -113σ² against -29σ² — so the two close on each other. At 11% they are still apart; at 12% the bands overlap and one study's answer could have come from either tree; at 20.5% the means cross, and above it a tree built at 3 measures lower than a tree built at 2.

Where three and two become one

Two trees, one built to obey Murray's law and one to obey Da Vinci's, are measured through the same fifty junctions with the same instrument. At twelve per cent of error on each radius the two answers overlap, and above twenty and a half the tree built at three measures lower than the tree built at two.

 The fork the cost chooses at a daughter ratio of 1: 37.47 and 37.47 degrees. Three ends held fixed, three weights fixed by the radii, and the branch point put where the total cost is least. At the optimal radius the pumping term is exactly half the upkeep term, so a segment's weight is its own cross-section and the three weights here are 1.5874, 1.0000, 1.0000. Minimising directly over the position — a 41×41 grid re-centred and shrunk 220 times, told nothing about any formula — puts the daughters at 37.4673° and 37.4673° from the parent's own forward direction, against the closed form's 37.4673° and 37.4673°.

The angle the cost chooses

Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.

 The three weights at an exponent of 2 and 3: one closes a triangle and one is a straight line. A branch point minimising a weighted sum of three lengths has an interior solution only when the three weights close a triangle, and the weight on a segment here is its own cross-section. At an exponent of 3 the three areas clear that condition by 0.4126, and the triangle they close is what the two fork angles are read off. At an exponent of 2 the parent's area is exactly the daughters' areas summed — that is what area conservation says — so the slack is -4.44e-16, the triangle collapses onto a line, and the fork closes to 0.0000 degrees. Leonardo's rule does not predict a different angle here; it predicts no angle.

A rule that predicts everything

Leonardo's rule says a fork conserves cross-section, and cross-section is exactly the weight the branch point is minimised against. So the three weights land on the boundary of the triangle inequality, the cosine comes out at one to the last bit, and the rule predicts no angle at all — and the free constant its own derivation leaves behind then walks the prediction across every angle a fork could have.

 What insisting on a fork angle costs: 43.3 degrees for one per cent of the network. The three ends are held where the optimum wants them, the branch point is moved everywhere inside them, and the cheapest network at each total angle is kept. The optimum sits at 74.93°. Everything within one per cent of the least cost runs 55.9° to 99.2° — a span of 43.3°, which read back as exponents covers 2.44 to 5.34 — and within a tenth of a per cent it still runs 13.6°. The prediction is steep in the exponent and the cost is nearly flat in the angle; they are the same curve read along its two axes.

An optimum too flat to reach

One per cent of a branching network's cost buys forty-three degrees of fork angle, covering exponents from 2.44 to 5.34, while the angle the theory predicts moves only fourteen and a half degrees across every daughter ratio there is. The prediction is steep and the cost is flat, and those are the same curve read along its two axes.

 The two trees this site draws, at 30° and 32° to a side, against the cost's 37.47° and 37.47°. Two trees of 63 segments each, 5 generations deep and 31 junctions apiece, with every junction's radii taken from r₀³ = r₁³ + r₂³ exactly and every junction's angle taken from a constant. Read as an exponent through cos(θ/2) = 2^(2/p − 1), the drawn angles say 2.5237 and 2.6239, in pictures whose widths are built at exactly 3. The cost that fixed those widths wants 37.47° and 37.47° at this daughter ratio, 74.93° in total, and the misses cost 0.573% and 0.292% of the network — which is why a fixed angle can sit in a figure about a minimisation and never look wrong.

The trees drawn at no angle

Two branching figures in these essays set every junction's radii from the cube law exactly and every junction's angle from a constant nobody derived. Read as exponents the drawn angles say 2.52 and 2.62, in pictures whose widths say exactly three — and at a lopsided fork the drawing puts a daughter thirty-four degrees from where the same cost puts it.

 What a swelling does to the exponent read from the informative band, by where the swelling is. 50 junctions of daughter ratio 0.6–1, built at exactly 3 and read with no random error, but with one or more radii measured fat. A parent read fat lowers the reading: 1% gives 2.864, 3% gives 2.631, 10% gives 2.075, and at 11.3% the tree reads Da Vinci's 2. Daughters read fat raise it: 1% gives 3.150, 3% gives 3.499; from 8% some junctions have a daughter measured wider than their parent, which no exponent balances, and the line stops. All three radii read fat by one factor return 3.000 at every swelling — the unswollen reading exactly.

A swelling at the fork

A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.

 Adding error to carry a fitted exponent back to none, at 12%. At 12% of error on every radius each replicate is refitted with more error added at four levels, and the dots are the means over 300 replicates, the tree at 3 above and the tree at 2 below. Uncorrected they read 2.035 and 1.676. Carried back to no error through the mean points, the tree at 3 reads 2.695 by a quadratic curve, 2.380 by a linear curve, 2.914 by a rational curve; the tree at 2 reads 1.961 by a quadratic one, 1.877 by a linear one, 1.982 by a rational one. The curve carried back is a choice the method does not make.

A correction that keeps the overlap

The duel between a tree built at Murray's exponent and one built at Da Vinci's ended by saying the displacement is the geometry, and that no better estimator removes it. Correcting every replicate by simulation-extrapolation removes 92 per cent of the tree at three's displacement at five per cent of error and 68 per cent at twelve, and the error at which the two means cross leaves the measured range altogether. It pays in spread — the corrected readings are twice as wide at twelve per cent — so the error at which the two trees' intervals overlap does not move. Of the duel's two numbers, the inversion was the estimator's and the overlap is the question's.

 Measured radii against the tips each branch carries, at 12% of error on every radius. One replicate of a 50-junction tree, 101 segments, every radius measured with 12% of relative error and drawn against the number of tips that segment carries. The count has no error in it, so the slope is not displaced: the tree built at 3 gives a slope of 0.3404, an exponent of 2.937, and the tree built at 2 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.

A count carries no error

Fitting r₀ᵏ = Σrᵢᵏ junction by junction puts a measured radius on both sides of every equation, and at twelve per cent of error a tree built at Murray's three and one built at Da Vinci's two stop being told apart, however the fit is corrected. Fit the same measured radii against the number of tips each branch carries instead — a count, which nobody measures with error — and the two trees read 2.996 and 1.997 at twelve per cent and 3.015 and 2.001 at thirty, never overlapping. The twelve per cent belonged to the junction fit, not to the tree. The count fails in its own way, and the way is stated.

 One tree sized for flow and for stress, with each branch 2^(−1/2) the length of its parent. The same symmetric tree, 8 generations deep, each generation's branches 2^(−1/2) the length of the one before and turned 30° at every fork, sized two ways and drawn to one trunk width. On the left each branch's radius cubed is proportional to the tips it feeds — Murray's flow rule — and every junction conserves r³. On the right each branch is sized so that the same load on every tip bends it to the same stress at its base, radius cubed proportional to the sum of its lever arms to its tips; its trunk junction conserves r to the power 1.967, its outermost junctions 1.349, against a deep-tree limit of 2.000. The two trees thin at different rates from the same trunk.

A cube law with a lever arm

Murray's exponent of three comes from moving fluid for the least work, and Da Vinci's two has had no derivation here, only the name of the mechanical answer. Size every branch so that the same wind on every tip bends it to the same stress, and a junction conserves r to the power 3/(1 + log₂(1/λ)), where λ is how much shorter each branch is than its parent. A crown that fills a plane gives exactly two; halving lengths gives one and a half; no shortening gives three. Murray's flow rule gives three at every λ, so the lengths of a tree's branches say which mechanism sized it.

 The fork angle Da Vinci's rule predicts, against the size of the fork, in one tree. A tree has one value of the constant Da Vinci's rule leaves free, so a fork's share of it falls as the square of the fork's size and the angle the rule predicts changes with size. For even forks it is 119.1° at a relative size of 0.1, 101.7° at a relative size of 0.5, 74.9° at a relative size of 1, 44.6° at a relative size of 2, 9.6° at a relative size of 10, where Murray's rule gives 74.93° at every size; for daughter ratio 0.5 it is 118.9° at a relative size of 0.1, 101.3° at a relative size of 0.5, 77.6° at a relative size of 1, 48.9° at a relative size of 2, 10.9° at a relative size of 10, where Murray's rule gives 77.58° at every size. From a total of 100° to 20° at an even fork is a factor of 8.93 in radius.

One constant for every fork

Da Vinci's rule leaves a free constant in the cost that sets a fork's angle, and running it over its range walks the predicted angle from nothing to 120 degrees, through Murray's 74.93. So no single fork can refute the rule. But the constant is one number for a whole tree, and a fork's share of it falls as the square of the fork's size — the constant is a radius axis. A tree spanning a factor of ten in radius must show forks from 29.4 degrees at its biggest to 111.6 at its smallest, a spread wider than one fork's flatness can hide, while Murray's angle is the same at every size.

 The exponent read from a tree that has lost tips, tips lost one at a time. The 50-junction tree built at 3 and at 2, 12% of error on every radius, 300 replicates of loss and measurement at each level. Shaded: the central 90% of the exponent read against the tips still there. Dotted: the mean read against every tip ever grown. Solid: the junction fit's mean over the junctions that survive. The count's two intervals are still apart with 70% of the tips gone, reading 2.366 and 1.576, and overlap by 80%. The junction fit's intervals overlap by 20%. At the heaviest loss drawn, 90%, the scars still read 2.998 and 1.997.

A count that has lost tips

Radii read against the tips each branch carries keep Murray's three apart from Da Vinci's two where junction fits cannot, because a count has no measurement error in it. A count of the tips a tree has is not a count of the tips it grew. Losing them lowers both trees' readings by one factor that belongs to the losses and not to the rule, so the count stops being right long before it stops telling the trees apart: on fifty junctions at twelve per cent of error, to seventy per cent of the tips lost one at a time, and only to about a quarter lost in whole limbs. Counting scars repairs single losses exactly. Nothing countable repairs a shed limb.

 The fork angle the transport cost predicts on a tree sized by stress, against the size of the fork. A tree sized so that equal loads on its tips bend every branch to one stress conserves rᵖ with p set by how much shorter each branch is than its parent, λ. On such a tree the cost that fixes a fork's angle carries one constant, entering a fork of size s as s^(2p − 6), so the predicted angle changes with size unless p is three. With the constant set so that a fork of relative size one opens at Murray's 74.9°: At λ = 0.707 (p = 2.000) an even fork opens at 119.1°, 111.6°, 74.9°, 29.4°, 9.6° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.74 (p = 2.091) an even fork opens at 114.5°, 106.4°, 74.9°, 41.1°, 30.1° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.794 (p = 2.250) an even fork opens at 106.3°, 97.6°, 74.9°, 53.9°, 46.4° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.87 (p = 2.498) an even fork opens at 92.7°, 85.5°, 74.9°, 66.0°, 61.5° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.95 (p = 2.793) an even fork opens at 78.4°, 76.6°, 74.9°, 73.4°, 72.2° at relative sizes of 0.1, 0.32, 1, 3.2 and 10. Murray's rule opens every fork at 74.9°.

Forks on a tree sized by stress

Da Vinci's rule predicts that a tree's forks open wider as they get smaller, because the constant in its angle cost is one number for a tree and enters each fork scaled by its size. A tree sized for equal bending stress conserves an exponent set by how much shorter each branch is than its parent, and on such a tree the same constant enters each fork as its radius to the power 2p − 6. The trend survives at every length ratio short of one and shrinks with it: 105 degrees a decade of radius for a crown filling a plane, 83 at a length ratio of 0.74, 53 for a crown filling a volume, 13 at 0.9. Only a crown shortening about as fast as a volume-filling one fans wider across a tenfold range than one fork's flatness can hide.

 One planar crown sized for a load on its tips and for the weight of its own wood. The same symmetric crown, 9 generations deep, each branch 2^(−1/2) the length of its parent and turned 30° at every fork, sized so that every branch is bent to one stress, drawn to one trunk width. On the left the load is on the tips and the trunk junction conserves r to the power 1.980; on the right the load is the weight of the wood, found by iterating the radii until they stop moving, and the trunk junction conserves r to the power 0.969. A crown sized for its own weight thins much faster from the trunk, because a branch's weight grows with the square of its radius.

A crown that carries its own wood

Sizing every branch so that equal loads on the tips bend it to one stress gives a crown filling a plane Da Vinci's exponent of two. Move the load onto the wood and the sizing becomes a fixed point, because a branch's load now depends on the radii being solved for. Under the wind on its wood a planar crown still conserves two, but only as a limit its trunk is two tenths short of at fifteen generations. Under its own weight it conserves one — radius rather than area, the stress-similarity law that radius goes as length squared — and a crown carrying leaves and wood reads the leaves' two near its twigs and the wood's one at its trunk, with the handover set by how much of the trunk's load the wood carries.

 The number of growing points after each season, for buds that wait no season, one, two, three or four. From one mature apex, each season every mature apex makes a new bud, and a bud branches only after it has waited its delay. With no delay the counts run 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and settle into growing by 2.0000 a season; with one season the counts run 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and settle into growing by 1.6180 a season; with two seasons the counts run 1, 2, 3, 4, 6, 9, 13, 19, 28, 41 and settle into growing by 1.4656 a season; with three seasons the counts run 1, 2, 3, 4, 5, 7, 10, 14, 19, 26 and settle into growing by 1.3803 a season; with four seasons the counts run 1, 2, 3, 4, 5, 6, 8, 11, 15, 20 and settle into growing by 1.3247 a season. On a logarithmic axis each settles into a straight line whose slope is its growth rate, the positive root of x^(d+1) = x^d + 1.

A count set by a delay

An L-system describes a plant and forbids nothing, because none of its parameters is anything a plant has. One branching grammar is the exception: a mature apex makes a new bud every season, and a bud waits d seasons before it branches. Its counts grow at the root of x^(d+1) = x^d + 1, a delay of one season gives Fibonacci's numbers and nothing else does, and the fourth count already separates a one-season wait from every longer one. So a Fibonacci count in a branching plant is a measurement of how long its buds wait. It is also a fragile one: if one bud in ten waits two seasons instead, eleven counts in a row come out Fibonacci's three times in a thousand.

 The rate a branching count grows at, against the chance a growing point dies. Every point dies with probability q each season and the survivors rewrite as before, so the expected counts obey x^(d+1) = (1 − q)·x^d + (1 − q)^(d+1). Substituting x = (1 − q)y returns the deathless equation exactly, which makes every line here straight: the rate is the deathless root multiplied by the survival. No delay runs from 2.0000 to one at q = 0.5000; one season runs from 1.6180 to one at q = 0.3820; two seasons runs from 1.4656 to one at q = 0.3177; three seasons runs from 1.3803 to one at q = 0.2755; four seasons runs from 1.3247 to one at q = 0.2451. Below the marked line a lineage shrinks.

A count that loses its growing points

The branching grammar behind the Fibonacci claim has no deaths in it, and a stem that loses shoots is the common case. Giving every growing point a chance q of dying each season leaves the counts a linear recurrence and does exactly one thing to it: the growth rate becomes the deathless root multiplied by 1 − q, at every delay and every death chance, to the last bit a double holds. So each waiting time has a death chance above which its lineage shrinks — a half with no wait, 0.3820 at one season, 0.2451 at four — and a longer wait tolerates less. What does not survive is the count itself: a plant losing one growing point in ten a season shows eight Fibonacci counts in a row one time in ten thousand, against one time in eight for a bud that occasionally waits an extra season.

 How many scars a lineage carries for every growing point still alive. Deaths arrive at q times the standing count and the count grows by x a season, so the scars settle at q/(x − 1) — the curves. The dots are the ratio the expected counts actually reach after eighty seasons, and they agree to within 1.0 per cent. A bud waiting no delay at q = 0.1 carries 0.1250; a bud waiting one season at q = 0.1 carries 0.2192; a bud waiting two seasons at q = 0.1 carries 0.3135; a bud waiting three seasons at q = 0.1 carries 0.4128. Each curve runs to infinity at its own threshold, where the living stop outgrowing the dead.

What a scar is worth

Counting the scars a dead shoot leaves does not put a branching count back on the sequence it would have had. A scar records a growing point and a dead growing point takes every branch it would have made, so living points plus scars reach 39.2 per cent of the deathless count after twenty seasons at one death in twenty, and 1.9 per cent at one in five — falling without limit rather than closing. What the scars restore is the other number. Scars per living point settle at q/(x − 1) exactly, so a rate with a scar share beside it recovers the death chance and then the waiting time, where a rate alone is reached by a one-season wait losing a tenth, a two-season wait losing 0.64 per cent and no wait at all losing 27.2 per cent.

 Sizing for equal bending and sizing for equal stress cross at one length ratio. Equal stress holds r³ against the sum of a load's arms and gives 3/(1 + ℓ); equal deflection holds r⁴ against the sum of the arms squared and gives 4/(1 + 2ℓ), with ℓ = log₂(1/λ). Setting them equal gives 3(1 + 2ℓ) = 4(1 + ℓ), whose only root is ℓ = 1/2 — the crown that fills a plane, λ = 0.707107 — and there both are exactly two. Below that ratio the stiffness rule reads the lower exponent of the two and above it the higher, so the two criteria size the same crown at one length ratio in the whole family and it is the one Da Vinci's rule names.

A crown sized for how far it bends

Stress is one criterion for sizing a branch and stiffness is another. Holding every branch to the same deflection as a share of its own length sizes r to the fourth against the sum of each load's arm squared, where equal stress sized r cubed against the arm, and the junctions of a deep crown then conserve 4/(1 + 2·log2(1/λ)). A single cantilever under its own weight comes out at radius as length to the three halves — McMahon's elastic similarity, fitted here rather than assumed — against the square that equal stress asks for. And the two criteria agree at exactly one length ratio out of the whole family: λ = 2 to the minus a half, the crown that fills a plane, where both give exactly two.

 Three ways of sizing a crown, and the exponent each one conserves. Murray's flow rule sizes r³ against the tips a branch feeds and conserves three at every length ratio, reading nothing of the lengths at all. Equal bending stress conserves 3/(1 + ℓ) and equal deflection 4/(1 + 2ℓ), where ℓ = log₂(1/λ). So an exponent measured on a tree names a rule only with a length ratio beside it, and even then not everywhere: the stress and stiffness curves meet at λ = 0.7071, the stiffness curve passes three at λ = 0.8909, and the stress curve reaches three only as the branches stop shortening.

Three rules, one exponent

A measured branching exponent is quoted as evidence for a sizing rule, and it cannot be. Murray's flow rule conserves three at every length ratio and reads nothing of the lengths at all; equal stress conserves 3/(1 + l) and equal deflection 4/(1 + 2l), where l is log2(1/lambda). So an exponent names a rule only with a length ratio beside it, and even then not everywhere: of ninety-six length ratios between 0.3 and 0.99, thirteen have two rules within five hundredths of each other at a precision of 0.05, in three bands with three different reasons — stress against stiffness where they cross at the planar crown, stiffness against flow where the stiffness curve passes three at 0.8909, and stress against flow only as the branches stop shortening.

 Each junction's exponent against its daughters' lengths, measured exactly and with a two per cent error in every radius. One thirteen-generation crown at a mean length ratio of 2^(−1/2), lengths spread by a half-width of 0.3 in log, its junctions five to ten generations above the tips — every third one drawn. Each junction appears twice, its exponent under equal stress and under equal bending, against the sum of its two daughters' log length ratios; the lines are the least-squares fits. Measured exactly, the stress exponents follow a slope of 0.298 and explain 62% of their scatter, the bending exponents a slope of 0.858 and 93%. With every radius read 2% wrong, the clouds swell — the stress scatter explained falls to 20% — and the slopes read 0.288 and 0.866. The error lands in the residual, and the slope is where the rule is.

The lengths that name the rule

A real crown has no single length ratio, and giving every fork a spread of daughter lengths does not blur what a sizing rule conserves: Murray's flow rule still conserves three at every junction, and the two mechanical rules keep their mean exponent, moved only as the square of the spread. What the spread adds is a second number. Each junction's exponent follows its daughters' summed log length with a slope of 0.30 under equal stress and 0.86 under equal bending at the planar crown, where the exponents are both two — and a two per cent error in every radius moves neither slope, while it swamps the scatter that looked like the obvious instrument.

 Which tilts buckling sizes and which bending sizes, deep in the crown and near its tips. Crowns at the planar length ratio, forks turned 20°, sized by the larger of buckling and bending, at five balance angles. For each, the upper bar runs from vertical to the widest tilt at which buckling sizes a branch seven or more generations above the tips, and on from the narrowest tilt bending sizes; the lower bar is the same reading within six generations of the tips. Balanced at 30°, buckling sizes the deep crown to 20° and bending from 40°, and near the tips buckling reaches 60°; balanced at 40°, buckling sizes the deep crown to 40° and bending from 60°, and near the tips buckling reaches 60°; balanced at 50°, buckling sizes the deep crown to 40° and bending from 60°, and near the tips buckling reaches 60°; balanced at 60°, buckling sizes the deep crown to 60° and bending from 80°, and near the tips buckling reaches 60°; balanced at 70°, buckling sizes the deep crown to 60° and bending from 80°, and near the tips buckling reaches 60°. In the deep crown the two never overlap at any generation, so the criterion is chosen by direction; near the tips the arms have not converged, the bending term is smaller, and the upright core is wider.

A crown that would rather not buckle

A column held below the load at which it buckles and a cantilever held to a fixed deflection need the same radius at every length, because both hold the bending stiffness against a load times a length squared — so the three halves of elastic similarity is also the buckling law, and a crown whose loads all run along its branches conserves the same exponent under either. Gravity does not run along branches. It divides by the cosine of each branch's tilt, so a buckling junction's exponent is set by the direction its parent points, nothing past level is sized at all, and a crown sized by the larger of the two criteria splits by direction into an upright core and a spreading shell whose boundary junctions conserve more than either rule gives.

 Fork angles on crowns sized by the larger of flow and stress, handing over at six different generations. Thirteen-generation crowns at a length ratio of 0.707, every branch sized by the larger of flow and stress, twigs at the radius the transport cost prefers, with the two radii equal at generations 3 to 8. Handing over at 3, the forks from the trunk open at 38, 51, 67, 75°; handing over at 4, the forks from the trunk open at 27, 38, 51, 67, 75°; handing over at 5, the forks from the trunk open at 19, 27, 37, 51, 67, 75°; handing over at 6, the forks from the trunk open at 13, 18, 26, 36, 50, 66, 75°; handing over at 7, the forks from the trunk open at 8, 12, 17, 24, 35, 49, 66, 75°; handing over at 8, the forks from the trunk open at 4, 6, 10, 15, 22, 32, 46, 65, 75°. Every fork from the handover outward opens at Murray's 74.9°. No fork on any of the six opens wider than that, and none turns back: the angle rises through the stress-sized generations and stops.

A trend that stops at Murray's angle

A crown sized by whichever of flow and bending stress asks for the thicker branch is sized by stress at its trunk end and by flow at its twigs, and the twigs fix the constant that was free in the fork-angle prediction: a twig at the radius the transport cost prefers spends exactly half its upkeep on pumping. On such a crown the fork angle does not change sign at the handover. It rises through every stress-sized generation, meets Murray's 74.93° at the handover and stays there, and no fork anywhere opens wider. The trend turns back only when the twigs are thinner than the cost wants — past a pumping share of (λ^(−2/3) − 1)/(1 − λ^(4/3)), 0.70 at the planar crown and closing on one half as branches stop shortening.

 Every pair of death chances that reproduces each count, for one plant. A plant with a wait of two seasons, apices dying at 0.05 and buds at 0.15: a rate of 1.3351, 0.3098 scars per living point and 0.2225 of its scars left by apices. Each line is every pair of chances at that wait reproducing one count; the dashed lines either side are the same count one per cent high and low. The rate's and the scar share's lines cross at the plant's own pair, at an angle of 11.6°, so a one per cent error lets the pair slide along them — apex chances from 0.004 to 0.097 and bud chances from 0.100 to 0.198. The line for scars sorted by kind crosses the rate's at 51.5°, and read with it the same error leaves 0.044 to 0.056 and 0.134 to 0.166.

Two ways to die, three things to count

Giving a branching plant's waiting buds a death chance of their own leaves its counts a linear recurrence, but breaks the collapse onto the survival: the rate becomes the apex survival times the root of y^(d+1) = y^d + r^d, where r is the bud survival over the apex survival. The one-chance reading then names the wrong waiting time on 171 of 477 plants with waits of two to four seasons, shorter when the buds are the fragile ones and longer when the apices are. The two chances are separable from a rate and a scar share, exactly — but the two counts' loci cross at eight to sixteen degrees, so a one per cent error lets the chances wander by a factor of two. A third count is owed, and it is the scars sorted by kind.

 Four hundred plants whose seasons are sometimes bad, each measured against the count the average season predicts. A two-season wait, and every season bad with chance 0.1, killing each growing point with chance 0.5; good seasons kill with 0.0556, so the chance averaged over seasons is 0.1. Four hundred sequences of forty seasons, each plant's living count divided by the count the independent model expects at 0.1. The dashed line at one is the expectation over every sequence of seasons; the mean of these four hundred is 1.073 at season forty, as near as a sample carried by its luckiest few plants comes. The solid line is the median plant, 0.540 of the expected count by season forty, and the band holds the middle ninety per cent, from 0.080 to 3.640.

A bad year does not average out

Let every growing point on a plant share one season's death chance, bad one season in ten and good otherwise, with the average held at 0.1. Averaged over every sequence of seasons the counts are exactly the independent model's. But no plant is an average over sequences: with bad years at 0.5 a plant settles on a rate of 1.3001 against the expected 1.3190, the median plant holds 54 per cent of the expected count by season forty, and the scar share never settles, because it is set by how many seasons ago the last bad year was — 1.35 the season after one, 0.17 twelve seasons on. So the reading of a plant's waiting time from a rate and a scar share gets worse the longer it runs: right for 57 per cent of plants over ten seasons, 22 per cent over eighty.

 One plant's scars dated by season, and the wait they name when the recurrence is run through its own seasons. A plant with a two-season wait, grown point by point for 20 seasons from one apex; each season is bad with chance 0.1, when each point dies with chance 0.5. Top: the share of living points that died each season, read from that season's scars, with the bad seasons marked. Bottom: the living count on a logarithmic scale, and the counts a wait of one, two and three seasons predicts when the recurrence is run from one apex through these same seasons' death shares. The two-season line follows the plant; dated, the scars name a wait of 2. Read as two totals — the rate after the first 5 seasons and scars over living points — the same plant names 1.

Scars with dates on them

A bad season shared by every growing point wrecks the reading of a branching plant by two totals: over eighty seasons the rate and the scar share name a two-season wait for 22 per cent of plants. Date the scars — by position along a shoot, by growth ring — and each season's death chance is read off its own scars, so the bad years stop being noise and become a known input. Running the branching recurrence through the plant's own seasons names the wait for 94 per cent of plants over twenty seasons whether bad years kill a tenth of the points or seven tenths, and for every plant over eighty. What a plant cannot read from its own scars is the climate: the averaged chance comes only as fast as seasons do.

 One plant's scars sorted into apices and buds and dated by season, and the kind of bad year its seasons show. A plant with a two-season wait, grown point by point for 20 seasons from one apex, apices dying at 0.05 and buds at 0.15 averaged over the climate: frost, bad years 5 times a good one. Each season has two bars, the share of living apices that died that season and the share of living buds; bad seasons (6, 10, 11, 18) are marked. In a frost the bud bars jump and the apex bars do not. Run through both series, the recurrence names a wait of 2; run through one share per season it names 2. The plant ends with 438 living points.

A frost and a drought in the scars

A branching plant whose buds and apices die at chances of their own, in a climate with bad years, leaves two kinds of scar in every season. Sorted by kind as well as dated, the scars give two shares a season, and the branching recurrence run through both names a two-season wait for 97 to 98 per cent of ten-season plants where one share a season names it for 81 to 83 — even when buds and apices die alike, because a small plant's deaths fall on one kind or the other by chance and the sorted series knows which. The three totals that the two chances were first thought to need fail under either kind of bad year, leaving the bud chance a tenth out however many seasons are read. And the sorted series can say what kind of bad year it had: a frost that strikes buds alone is told from a drought that strikes both at nine plants in ten over forty seasons, and one frost season is seen once the plant carries a few hundred buds.

 A crown sized against buckling on clamped bases, coloured by the share of that load each branch keeps on the base its parent really gives. A symmetric crown 9 generations deep, every branch 2^(−1/2) of its parent's length, turned 20° at every fork, with equal loads on the tips, sized against buckling as if every branch were clamped. Each branch is coloured by the share of its clamped buckling load it keeps once its base is the tip of a parent that bends, the parent alone: from 0.296 to 0.795 outside the trunk. A daughter turning back toward vertical off a steep parent keeps least, one turning outward most. Branches pointing level or down carry no compression and are drawn faint.

A base that gives

Every branch sized against buckling was a column clamped to a parent that does not move. A parent bends, and a column on a base that turns keeps only a share of its clamped load — the root of φ·tan φ = kL/EI — which no stiffening of the column itself lifts past the load at which a rigid rod on that base would tip. On the crown sized as if clamped, that share is 0.561 at every branch when the loads run along them, falls from the ground to 0.447 when the whole path is counted, and under gravity is a closed form in the turn from a parent's tilt to its daughter's: 0.795 for a daughter turning outward, 0.296 for one turning back upright off a steep parent. Resized for the bases they really have, the crown's junctions alternate about two from generation to generation, 226 branches of a thirteen-generation crown have no radius that holds them, and the cone of directions that divided buckling from bending no longer divides them.

 A crown whose parents are sized for the bases their daughters need, coloured by the term that sizes each branch. A symmetric crown 9 generations deep, every branch 2^(−1/2) of its parent's length, turned 20° at every fork, under gravity, sized from the trunk outward by the largest of three terms: the radius its own column needs on the base its parent gives, the base term its daughters demand at a margin of 2.5 over the load at which a rigid rod would tip on that base, and the bending rule, balanced against buckling on the trunk at 50°. The base term sizes 249 branches, the column term 238 and bending 24; no branch is left without a radius.

A parent sized for its daughters

A branch whose compression times its length exceeds what its parent's tip could hold as a rigid rod cannot be saved by thickening the branch; the requirement is the parent's. Size every parent for the base its daughters need, at a stated margin over that ceiling, and nothing in a crown is left without a radius. With every load along its branches the junctions stop alternating at one margin, 2.043 at the planar ratio, and conserve exactly two above it. Under gravity a junction sized this way conserves the clamped crown's closed form read one fork further up. Joined with bending, the cone of directions comes back, wider. And the cheapest crown sits below the margin at which the alternation stops, at about 1.6 to 1.7, so the zigzag is not waste.

 How often a late window of dated, sorted scars names a plant's wait, against the early seasons lost: a wait of 3. A wait of 3, 2000 finite plants, apices dying at 0.05 and buds at 0.15 on average; drought, bad years 5 times a good one. Each plant's first seasons are lost and the next 10 read, dated and sorted, from four starts at the window's first season. True ages: 90.4%, 92.2%, 95.8%, 97.5%, 99.4%, 99.9%, 100.0%. Steady-growth ages: 90.4%, 78.9%, 91.4%, 94.1%, 98.4%, 99.9%, 100.0%. Living total split: 10.8%, 77.0%, 87.8%, 93.0%, 98.2%, 100.0%, 100.0%. No history: 90.4%, 3.0%, 9.6%, 4.9%, 2.0%, 0.1%, 0.0%, after 0, 2, 4, 6, 10, 20, 30 seasons lost. From the first season, one dated share a season names the wait for 52.3% and the sorted series for 90.4% — the two dashed lines. The median plant at the window's start holds 1, 3, 4, 7, 15, 133, 1147 points.

The seasons a plant no longer shows

A branching plant found in the field has grown over its oldest scars. Read dated and sorted from a window of ten seasons that starts late, it names its wait better than from its first season, not worse — a two-season wait for every plant once ten seasons are lost, and a three-season wait for every one once thirty are — because the seasons it keeps are large ones. What the window's start must supply is the buds counted as buds: treat every living point as an apex and the reading names almost nothing. Split the buds by age as steady growth would, and the only loss that costs is a small one, on a young plant that is not yet growing steadily. One share a season catches up in a calm climate and never under bad years. And the frost test, which waited for buds, gains.

 How often a late window names a plant's wait when its youth grew under another climate. 2000 plants with a wait of two seasons and 2000 with a wait of three, grown 20 seasons before a window of 10 is read, dated and sorted, from the steady-growth start; the climate before the change is one of four youths and after it the present drought, apices dying at 0.05 and buds at 0.15. More bad years, a wait of 2: 100.0%, 100.0%, 100.0%, 100.0%, 100.0%, 100.0%. Dear youth, a wait of 2: 100.0%, 100.0%, 100.0%, 100.0%, 100.0%, 100.0%. Kind youth, a wait of 2: 100.0%, 100.0%, 100.0%, 100.0%, 100.0%, 100.0%. Bud-killing youth, a wait of 2: 100.0%, 100.0%, 100.0%, 100.0%, 100.0%, 100.0%. More bad years, a wait of 3: 99.9%, 99.9%, 100.0%, 99.9%, 99.9%, 99.9%. Dear youth, a wait of 3: 97.4%, 97.8%, 98.2%, 98.4%, 99.2%, 100.0%. Kind youth, a wait of 3: 100.0%, 100.0%, 100.0%, 100.0%, 100.0%, 100.0%. Bud-killing youth, a wait of 3: 97.6%, 97.9%, 98.8%, 98.9%, 99.0%, 99.9%, with the change 0, 1, 2, 3, 6, 10 seasons before the window. Steady plants: 100.0% and 100.0%. Every wait of two is named; a wait of three is missed on at most 2.6% of plants, those whose dear or bud-killing youth ended as the window opened.

A plant's buds forget the climate that grew them

A branching plant read from a late window was named right when the seasons it had lost looked like the seasons it kept. Grown under a harsher, kinder or more erratic youth and read after the climate changed, it still is: a wait of two seasons for every plant, a wait of three for 97 to 100 per cent. The hoped-for sign of the change — buds and apices at the window's start in a mix the window's own survivals would not produce — barely exists, because a steadily growing plant holds about the same number of buds per apex whatever it dies of. Death chances that differ tenfold move that ratio by a few per cent, the plant forgets a change within two turns of its wait, and what a changed youth does leave is the plant's size.

 One plant whose wait lengthens from two seasons to three once it reaches a hundred living points, and the windows it can be read from. A branching plant under the present climate whose buds wait two seasons until its living total first reaches 100, at season 15, and three seasons after. Its living total by season, with the change marked and a ten-season window opening at season 10; the change falls in season 5 of the window. Read dated and sorted from the steady-growth start, windows opening at seasons 0, 5, 10, 15, 20 name waits of 2, 2, 2, 3, 3 seasons; this one names 2.

A late window names the wait it mostly saw

A branching plant read from a late window names its buds' wait reliably, and its age structure forgets the climate it grew under. Every plant there kept one wait for life. Give a plant a wait that lengthens from two seasons to three once it reaches a stated size, and the window names the wait that held for most of the ten seasons it covers: the new wait if the change fell in its first three seasons or before, the old one from its sixth season on, and never a third. So across plants, read at their sizes, a wait that lengthens with size shows as one — the reading turns at a quarter to a half of the size the wait changed at. The age structure, which forgets a climate, remembers a wait that shortened for two seasons and one that lengthened hardly at all.

 A crown sized branch by branch for its loads, and the shape it buckles in as one frame. An axial crown seven generations deep, every parent sized for the base its daughters need at a margin of 1.7 over the rigid-rod ceiling and every branch to carry its own load on its parent's spring, assembled as one planar frame of beam-column elements and buckled. The frame buckles at 0.205 of the loads it was sized for. The lowest mode, drawn over the crown, leans the trunk and the first forks and carries the crown above them over: the trunk holds 24 per cent of the mode's strain energy and the first three generations 68 per cent. The branch the per-branch model holds at the smallest margin is in generation 1 and carries 9.4 per cent.

A crown sized branch by branch buckles as one at a fifth of its load

Every branch of a crown can be sized to carry its own load on the spring its parent's tip gives it, and every parent for the base its daughters need — and the crown, assembled as one frame, still buckles at a fifth to a third of the loads it was sized for. The spring stood in for the parent; it did not stand in for the crown above a branch, which leans when the branch leans and hangs its loads further out. The mode lives at the bottom, in the trunk and the first forks, not in the branch the spring model marks weakest. And the cheapest margin moves: branch by branch it was 1.7; held as one frame it is 4, on a floor so flat that 1.7 costs under two per cent more.

 Two lopsided crowns sized by the larger of flow and stress, drawn at the angles their radii predict. Two crowns of 9 orders whose every fork sends a thick daughter one order down and a thin daughter 2 orders down, so the crown has 89 twigs; each branch is 0.794 of its parent's length times the shortening of its order, and each takes the larger of the radius equal bending stress asks for and the radius flow asks for, the two equal at order 4. Every fork is drawn with each daughter at the angle the transport cost predicts from that fork's three radii and tips. On the left the twigs are the radii the cost itself prefers, and no daughter anywhere opens wider than the angle Murray's law gives its own fork. On the right the twigs spend twice their upkeep on pumping: the thin daughters open furthest past Murray's angle, by 12.5°, at the fork one order inside the handover, and the thick daughters furthest, by 10.4°, at a fork 1 order nearer the trunk. Branches sized by stress and branches sized by flow are drawn in two colours.

Two daughters cross Murray's angle together

On a crown sized by the larger of flow and stress whose every fork is lopsided, the thin daughter's subtree a few orders shallower than its sister's, the handover stops being one generation and becomes a band of forks one wider than the offset. Every fork past the band still opens each daughter at its own Murray angle, to a ten-trillionth of a degree. The thin daughter never overshoots alone: at the first fork whose daughters are both sized by flow, only the parent's weight differs from Murray's, and one side of a weight triangle passes its Murray length at a single value, so both daughters cross together, at a twig share within two hundredths of the even crown's. Past it they peak apart, the thin one by more.

 Two crowns whose forks are lopsided by different amounts, drawn at the angles their radii predict. Two crowns of 9 orders whose every fork sends its thin daughter's subtree nought to three orders shallower than its sister's, the offset drawn at each fork; 76 twigs, each branch 0.794 of its parent's length per order, sized by the larger of flow and stress with the two equal at order 4 for the median branch of that order. Every daughter is drawn at the angle the transport cost predicts from its fork's three radii. On the left the twigs are the radii the cost prefers and no daughter opens past its Murray angle; on the right they spend twice their upkeep on pumping. This crown first turns back at a twig share of 0.512, at a fork whose parent is 1.008 times as thick as flow alone asks.

Mixed forks turn back at the cost's own share

On a crown sized by the larger of flow and stress, a fork whose two daughters are sized by flow opens both past their Murray angles exactly when the twigs spend more than q⁴/(1 + q²) of their upkeep on pumping, where q is how much thicker than flow its parent is — a closed form that is the even crown's threshold and every one-offset crown's at once. A crown lopsided by one amount at every fork has one q at its handover and a margin above the cost's own one half: 0.64 at a length ratio of 0.794. A crown lopsided by different amounts has a parent at every q, some barely thicker than flow, and turns back at 0.500 to 0.514. A branch's weight is its cost per length, the cost is flat at its own radius, and a slightly thickened parent with slightly thin twigs is lighter than its daughters expect.

 Read at an opening of a degree, a blend turns the mixed crowns back later than the larger of flow and stress does. Twelve mixed crowns of fourteen orders and the even crown, every branch sized by the blend of flow and stress radii at the exponent on the axis, and in the last column by the larger of the two. For each, the twig share at which some daughter first opens more than 1° past its Murray angle. The bars are the twelve mixed crowns' range and the rings the even crown. At k = 2 the mixed crowns turn back at 1.122 to 1.151, at k = 4 at 0.817 to 0.821, at k = 8 at 0.706 to 0.754, and under the larger of the two at 0.603 to 0.672; the even crown at 1.084, 1.061, 0.943 and 0.724. The faint line is the same crowns read strictly, whose mean falls to 0.5014 at k = 8.

A blend turns the twigs back first

Size every branch of a mixed crown by a blend of what flow and stress ask, rather than by the larger, and no branch is exactly flow-sized any more. Read strictly, that removes every margin the crowns had: the forks of two twigs open past Murray's angles first, on a closed form of their own, and by a blend exponent of eight every crown does so within two thousandths of the cost's own one half, the even crown's 0.643 included. But those openings are thousandths of a degree. Read at a degree a protractor could see, a blend restores a margin at every exponent tried, 1.12 at two, 0.82 at four, 0.71 to 0.75 at eight, against 0.60 to 0.67 for the larger, and the fork that opens first climbs from the twigs to the handover as the blend sharpens.

 The highest and lowest factor a loss of a given share can put on a tip count, by which tips it takes. The 50-junction tree with no error on any radius. For each share of its 51 tips, a search over which tips make up that share for the set that reads the factor highest and the set that reads it lowest, by single swaps from 20 starts: 10%, 0.954 to 1.040; 20%, 0.906 to 1.062; 30%, 0.863 to 1.093; 40%, 0.812 to 1.121; 50%, 0.743 to 1.173; 60%, 0.684 to 1.162; 70%, 0.599 to 1.176; 80%, 0.509 to 1.137; 90%, 0.358 to 0.882. A factor above one reads both rules high. The line is the mean factor when every tip is equally likely to go. The true range is at least this wide, since a search can stop short of the extreme.

A loss that picks its tips can read high

A tree that has lost tips at random reads its branching exponent low, by one factor shared by every rule. Let the loss choose which tips go and the factor is still shared — the ratio of three to two survives any choice — but it is no longer below one. On fifty junctions with half the tips gone, the choice of which half moves Murray's tree from 2.23 to 3.52, and past about half the tips a single tree's reading can no longer say which rule built it. The ways a crown plausibly chooses stay within a few hundredths of a fair loss, all of them low, and counting scars repairs every one.

 One plant whose buds wait three seasons with a chance of 0.5 and two seasons otherwise, read from five windows. A branching plant under the present climate, every bud its surviving apices make waiting three seasons with probability 0.5 and two seasons otherwise. Its living total by season, with ten-season windows opening at seasons 0, 5, 10, 15, 20 marked along the foot; read dated and sorted from the steady-growth start, they name waits of 3, 3, 3, 2, 2 seasons. Every season on this plant holds buds of both waits.

A plant with two waits is named for the nearer

A late window reads a branching plant as having one wait. Give every bud its own — two seasons or three, by a stated share — and the window names the wait whose growth the mixture's is nearer: two below a share of 0.47, three above, never an average and no third wait more often than a pure plant shows. The buds per apex, which caught a shortened wait, cannot see the mixture on any plant large enough to matter. What does see it is reading the same plant through several windows: near the turn, nine plants in ten are named both two and three.

 A crown of 63 branches after 8 steps of moving wood towards its lowest buckling mode. Each branch drawn to its radius, scaled so the whole crown just holds the loads it was sized for. At step 8 it takes 43.76 of wood where the uniform repair takes 44.89 and the best reached takes 43.76.

Moving the wood saves almost nothing

A crown sized branch by branch buckles as one frame at a fifth of its load, and the repair so far has been to thicken every branch alike. Its lowest mode lives in the trunk and the first forks, so thickening a twig looks like waste. Moving each branch's wood to match its share of the mode's strain energy tests that, and it finds one crown, the same from every starting margin to four figures. From the crown that margin of 1.7 had made cheapest it saves 2.5 per cent, and from a margin of four, one. A twig holds very little of the mode's energy and also very little of the wood. The best crown does not keep one exponent: from about two and a half at the trunk's fork it falls to about one and a quarter at the twigs.

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