A plant's buds forget the climate that grew them

The seasons a plant no longer shows read a branching plant’s dated, sorted scars from a window that starts late — the stretch a field worker can still date on a plant that has grown over its oldest seasons — and named the plant’s wait for every plant once the window’s opening buds were split by age the way steady growth at the window’s own survivals would split them. That start is an assumption: that the plant has been growing steadily under the climate the window shows.

Every plant in that essay grew under one climate, so the assumption held by construction. That essay’s closing question was the case where it does not: a plant that spent its youth under a run of droughts, or a kinder decade, and is read after the climate changed. Would the steady-growth start misplace the buds’ ages and cost the reading its wait? And could the disagreement itself be read — a plant whose buds and apices at the window’s start are not in the proportion its window’s survivals would produce is a plant whose past was different?

Four youths #

The plant is the same plant: grown point by point from one apex, each mature apex making a bud a season, each bud maturing after the wait, mature apices dying at 0.05 a season on average and buds at 0.15, one season in ten a bad one with both chances five times a good year’s. That is the present climate, and every window is read under it.

Before the change the plant grows under one of four youths. A variable youth has bad years four times as often, with the averaged chances held — the same mean, a rougher ride. A dear youth doubles both averaged chances. A kind youth cuts both to under a third. A bud-killing youth more than doubles the bud chance and holds the apex’s. The dear and bud-killing youths have bad years three times a good one rather than five, because a doubled bud chance at five times would put a bad year’s chance of a bud dying past one.

Each plant grows twenty seasons before a ten-season window is read, and the climate changes 0, 1, 2, 3, 6 or 10 seasons before the window opens. Two thousand plants are grown for each youth, change and wait, and only the plants alive at the window’s end are read, as only those could be found.

The wait is still named #

A wait of two seasons is named for every plant read, under every youth, at every distance of the change. A wait of three is harder, as it was before, and is named for 97.4 per cent of plants whose dear youth ended as the window opened, 97.6 for a bud-killing youth, and from 98 to 100 per cent once the change lies two or more seasons back. The variable and kind youths cost nothing. Started from the buds’ true ages instead — the bound no field reading has — the dear youth’s three-season plants are named for 99.4 per cent, so the steady-growth start costs two points at worst.

So the first half of the question has a plain answer. A changed climate before the window does not cost the late reading its wait. That is surprising only until one asks how far the steady-growth start can be wrong.

How far the start puts the buds #

The steady-growth start does misplace the buds after a changed youth, and by a measurable amount. Counted as the share of a plant’s opening buds the split puts in the wrong age — half the summed difference between the true shares of each age and the split’s — a steady plant with a wait of three has 4.5 per cent of its buds misplaced, which is nothing but the scatter of finite plants about the steady mix. One whose dear youth ended as the window opened has 13.4 per cent misplaced, and one whose bud-killing youth ended then, 14.8: too many of the youngest buds and too few of the oldest, a youth’s slower growth still showing in the ages. Three seasons after the change the misplaced share is 9; ten seasons after, 6. The kind youth, whose plants are large, misplaces 2 to 4 per cent.

That is where the two points of the three-season reading go, and why they come back as the change recedes. It is also small. A seventh of the buds in the wrong age is a start the window’s ten seasons of dated deaths can recover from, which is the same tolerance the late reading showed when the buds were not split at all and only their kind was known.

How many buds per apex a plant holds #

A plant growing steadily under any climate settles into a fixed mix of mature apices and buds of each age, and the ratio of buds to apices in that mix is $\sum_{j<d} s_a s_b^{\,j}/\rho^{\,j+1}$, where $s_a$ and $s_b$ are an apex’s and a bud’s chance of surviving a season and $\rho$ is the plant’s growth factor, the root of $\rho^{d+1} = s_a\rho^{d} + s_a s_b^{\,d}$ that the wait itself sets.

That ratio hardly moves. For a wait of two seasons it runs from 1.128 to 1.197 across the whole grid — apex death chances from 0.015 to 0.15, bud death chances from 0.045 to 0.35, tenfold and eightfold ranges — and sits at 1.165 under the present climate. The dear youth’s survivals give 1.182, the kind youth’s 1.153 and the bud-killing youth’s 1.194. For a wait of three the whole grid spans 1.594 to 1.631, and the present climate gives 1.630.

Why the mix does not care #

The reason is in the formula. A bud that survives better makes the numerator larger, since more buds live to be counted; but a plant whose buds survive better also grows faster, so $\rho$ rises, and each term is divided by a power of it. The two move together. A harsher climate kills more buds and slows the plant by nearly the same factor, and the share of the plant that is buds stays where it was.

What the ratio does depend on is the wait. Two seasons of buds per apex make a ratio near 1.16, three near 1.63, because a longer wait holds more ages of bud on the plant at once. That is why the steady-growth start can name the wait at all: it splits the buds by the candidate wait’s own proportions, and those proportions differ far more between waits than between climates.

A change, and how long it is remembered #

The steady ratio is one thing; a plant just after a change is another, since its buds were made under one climate and are dying under the next — the situation a bad year that does not average out described for a single season, stretched over a decade. Followed without chance — the recurrence at each climate’s mean survivals, run until the youth’s mix has settled and then switched — the plant’s buds per apex depart from the present’s steady ratio by at most a few per cent and come back within two turns of the wait. With a wait of two seasons, a bud-killing youth leaves the ratio 2.5 per cent high the season of the change and 6.1 per cent high the season after, then 2.5 per cent low, 1.0 low, 1.0 high and level by the fifth season. A dear youth’s departures are half as large, a kind youth’s smaller and the other way.

The departure is largest one season after the change rather than at it, because the first season under the new climate kills buds made under the old one at the new rate while the apices those buds would have become have not yet appeared. Then the plant rings, once a wait, and settles.

Can the window see the change? #

The second half of the question was whether that departure can be read. A window gives its own survivals, from which the steady ratio follows, and it gives the buds and apices living at its start. A plant whose opening ratio is far from the steady one is a candidate for a changed past.

The test is set the obvious way: a plant is flagged when the log of its opening ratio over the steady ratio falls outside the middle ninety per cent of steady plants read the same way. On steady plants it flags ten per cent, by construction.

Against steady plants of the same age it looks as if it works. A dear youth that ended as the window opened is flagged on 37 per cent of plants with a wait of two, and a bud-killing youth on 35; the rate falls to 15 and 16 per cent ten seasons later. But plants with a dear youth are small — a seventh of a steady plant’s size at the window’s start — and a small plant’s ratio scatters more whatever its past, because a dozen buds is a noisier fraction than a hundred.

Read against steady plants of the same size instead — steady plants read younger, at the age whose median size matches — the same dear youth is flagged on 18.5 per cent, the bud-killing youth on 21.7, falling to 13 per cent three seasons after the change and to 9 to 12 per cent from six on. Twice the null rate at best, for one or two seasons, and then nothing. The variable youth, which changed only how often the bad years came, is flagged at about the null rate, 4 to 11 per cent.

What the scatter looks like #

Drawn for the plants whose dear youth ended the season the window opened, the two distributions share a centre — medians of 0.011 and close to nought — and differ only in their tails: the changed plants have a few more far to the right, the departure the expected plant showed in its first seasons. Moved back by the dial, the tails close: six seasons after the change, 9.3 per cent of the changed plants fall outside the steady plants’ middle ninety per cent.

So a plant’s buds do record a change, but only for about a wait and only in a minority of plants, and a reading that flagged them would flag almost as many steady plants beside them. The disagreement the question hoped to read is real for a season or two and then gone.

What a changed youth does leave #

The youth is not lost without trace. It is written in the plant’s size. A steady plant holds a median of 464 living points at the twentieth season. One whose dear youth ended as the window opened holds 64, and one whose bud-killing youth ended then holds 79; a kind youth leaves 1,712. A variable youth, which kept the averages, leaves 459, indistinguishable from steady.

That record needs one more number to read: the plant’s age. It is the same lesson what a scar is worth drew from a single scar — a count says little until something dates it. A worker who knows a plant has grown twenty seasons and finds 64 living points knows its past was harsh, whatever its buds say. A worker who has lost the plant’s first seasons has usually lost its age with them, and a small plant is equally well a young plant under the present climate — which is the trade the late window already accepted when it gave up the first seasons.

A rougher youth leaves nothing at all #

The variable youth is the limiting case. Its bad years came four times as often, but each was milder in proportion, so an apex and a bud died at the same average rates as under the present climate. Everything a window or a ruler could read agrees that nothing happened: the wait is named for 99.9 to 100 per cent of plants, the misplaced share of buds is the steady plant’s 2.4 and 4.5 per cent, the opening ratio is flagged at about the null rate, and the median plant holds 449 to 459 living points against a steady plant’s 464. A decade of erratic weather with the same mean is, to this plant, the same decade. Only a reading that sorted the bad years themselves — the frost test of the sorted series — could see that they had come more often, and only inside the window.

Why the reading survived #

Put the pieces together and the robustness of the late reading is no accident. The wait is named from how the window’s living counts follow from its start through its own survivals, and the start enters through the mix of buds and apices. That mix is set by the wait and barely by the climate, so a start computed from the wrong climate is nearly the right start. Where it is wrong — a season or two after a change — it is wrong by a few per cent, and the counts that follow over ten seasons swamp it.

A frost and a drought in the scars found that sorting the deaths by kind is what lets a plant tell its own bad years apart. The same sorted series is what makes a past climate irrelevant: the window’s survivals are read season by season, so the window carries its own climate with it and needs nothing from before but a mix that does not depend on the past.

Survival is not all a climate changes #

That a real plant’s mix of buds and apices is set by survival alone. A plant that changes its branching as it ages — a longer wait on a large plant, buds that stay dormant for several seasons under stress — holds a mix that records something, and nothing here grows one. Nor that a climate acts on a plant only through the two death chances; a drought that slowed growth without killing would change the wait itself, which the steady-growth start assumes is fixed.

Every plant here is the same model with the same present climate, read over ten seasons after twenty, and the model is a rule chosen to make the arithmetic plain — a description rather than an explanation of any plant’s branching. A shorter window or a younger plant would make the first seasons after a change a larger share of what is read.

Readings that would bring the past back #

A youth under which the steady-growth start names the true wait for under 95 per cent of plants. A survival pair on the grid whose steady ratio of buds to apices is more than a tenth from the present climate’s at the same wait. A changed youth flagged against steady plants of its own size at more than a quarter of plants in the first seasons after the change, or more than fifteen per cent three or more seasons after it.

The wait survives, and the youth is in the size #

A branching plant whose youth grew under a harsher, kinder or more erratic climate is read from a late window as well as a steady one: a wait of two named for every plant, a wait of three for 97 to 100 per cent. The reason is that a steadily growing plant holds nearly the same buds per apex whatever it dies of, because better survival grows the plant faster in the same proportion; tenfold death chances move the ratio by a few per cent. A change leaves a departure of a few per cent that rings out within two turns of the wait, which a test against steady plants of the same size catches on about twice the null rate at best. The youth is recorded in the plant’s size, which only its age can read.

Still open: a plant that changes its own wait #

Every plant here keeps one wait for life, and that is what makes its mix of buds and apices so stable. A plant whose wait lengthens as it grows — a bud on a large plant taking a season longer to mature than a bud on a young one, as apical dominance strengthens — holds a mix that changes with its size rather than its climate, and the steady-growth start would then split the window’s buds by the wrong proportions on every plant, not only the ones whose climate changed. The next measurement grows plants whose wait steps from two seasons to three at a stated size, reads them from a late window, and asks whether the reading names the wait the plant has now, the wait it had, or neither, and whether the age structure, which forgets a climate, remembers a change of wait.