The seasons a plant no longer shows

Every reading of a branching plant so far has started at the plant’s first season. A count set by a delay grew its plants from one apex; scars with dates on them ran the branching recurrence through the plant’s own seasons from that one apex forward; and a frost and a drought in the scars sorted each season’s deaths into buds and apices and named a two-season wait for 97 to 98 per cent of ten-season plants, against 81 to 83 for one share a season.

A plant in the field does not keep its first season. Its oldest scars are grown over by bark or shed with the shoots that carried them, and what a worker can date and sort is a recent stretch of seasons on a plant that is already large. The last essay ended by asking what that costs: how many early seasons can be lost before the wait is named no better than by one share a season from the start, whether the living counts at the window’s start can stand in for the lost history, and whether the frost test — which already had to wait for the plant to grow — loses anything.

The reading from the first season #

The benchmark is the one the last essay set. Its plants are finite, grown point by point from one apex: mature apices die at 0.05 a season on average and buds at 0.15, one season in ten is bad, and in a drought both kinds’ chances are five times their good-year values that season while in a frost only the buds’ are.

Read from the first season, the dated and sorted series names the wait for nearly every plant over ten seasons and the single share a season lags by about fifteen points. At a three-season wait the lag is wider: the sorted series names it for 90.4 per cent of ten-season plants in a drought, one share a season for 52.3. Those two numbers are what a late window has to be measured against.

What the window’s start has to supply #

A window that starts at season $L$ still gives each season’s living apices and buds and each kind’s deaths — everything the sorted reading used — and it still gives the living counts at its own start. What it does not give is how old the buds at the start are. The recurrence needs that: a bud one season from maturing and a bud two seasons from it contribute to different future seasons, and the wait is exactly the question of how long a bud takes.

So the reading needs a start, and four are measured. The true ages split the starting buds as the plant really holds them; no field reading has this, and it is the bound. The steady-growth ages split them as a plant growing steadily at the candidate wait would, age $i$ in proportion to $(s_b/\rho)^i$, with $s_a$ and $s_b$ the window’s mean survivals and $\rho$ the root of $\rho^{d+1} = s_a\rho^d + s_a s_b^d$ — each candidate wait bringing its own split. The living total split does the same to the whole living count, not knowing which points are buds. And no history treats every point living at the window’s start as a mature apex. Each start is run forward through the window’s own two survivals a season, and the wait whose predicted counts best match the plant’s is named, by the same deviance the dated reading used.

Losing seasons makes the reading better #

The answer to the first question is that no number of lost seasons brings the reading down to one share a season from the start, and most losses make it better. With a three-season wait in a drought and the steady-growth start, the late window names the wait for 91.4 per cent of plants after four seasons lost, 94.1 after six, 98.4 after ten and 99.9 after twenty — past the 90.4 the plant’s own first ten seasons gave. By thirty seasons lost it names it for every plant. With a two-season wait the loss barely registers: 97.4 per cent from the start, 98.3 after four lost, 100 from ten on.

The reason is the size of the plant. At a three-season wait the median plant holds one point at its first season, seven after six seasons, 133 after twenty and over a thousand after thirty, and the recurrence is a statement about counts. On a plant of one to fifteen points every season’s deaths are a handful, and a wait of two and a wait of three predict counts that differ by less than their own noise. On a plant of a thousand, one season’s counts pin the survivals to a few parts in a hundred, and the candidates’ predicted counts diverge within a few seasons. The early seasons were never the informative ones. They were the ones in which the plant was too small to say anything.

The one loss that costs #

The exception is a small loss. Two seasons lost at a three-season wait bring the steady-growth start down to 78.9 per cent, below the 90.4 from the first season — the only point on the curve that is worse than keeping everything. The true ages give 92.2 at the same loss, so the cost is not in the window. It is in the start.

A young plant is not growing steadily. One season after it began, every bud it holds was made that season, and the steady-growth split puts 47 per cent of its buds at one season old, 32 at two and 22 at three: 53 per cent of the buds in the wrong age. After two seasons it has buds of one and two seasons and none of three, and the split still misplaces 27 per cent. The misplaced share falls to 17 per cent after six seasons, 10.6 after ten and 4.6 after twenty, as the plant’s own history settles toward the structure steady growth describes — which is also the structure the recurrence settles toward, so that by then the lost seasons carry no information the start has left out.

So the steady-growth start is a statement that the plant has been growing long enough for its age structure to forget its beginning. That is true of any plant large enough to be worth reading and false of a plant a few seasons old, and the reading’s one dip sits exactly where the statement fails. The living total split, which assumes steady growth for the apices too, fails the same way and harder on the youngest plants: it splits a single apex into a fraction of an apex and fractions of buds, and names the three-season wait for 10.8 per cent of plants read from their first season.

A shorter wait forgets its beginning sooner #

The dip belongs to the three-season wait, and the two-season wait shows why.

At a two-season wait the steady-growth start names the wait for 97.4 per cent of plants read from the first season and 97.4 after two seasons lost — no dip at all — then 98.3 after four and 99.4 after six. The true ages sit a point above it throughout, and the two meet at 100 per cent from ten seasons lost. The young plant’s age structure is still wrong for the split, but less wrong: after two seasons it holds 55.9 per cent of its buds at one season old against the split’s 61.2, a misplaced share of 16.4 per cent where the three-season plant’s was 26.9. A shorter wait has fewer ages to confuse and a steady structure that is reached in fewer seasons, since the transient a plant starts with dies away over a few multiples of the wait.

That is the general form of the answer to the first question. The lost seasons cost something only while the plant still remembers its beginning, and a plant remembers its beginning for a number of seasons set by its wait — the longer each bud waits, the longer the first apex’s single lineage stays visible in the ages of the buds. The wait being read sets how young a plant can be before its window can be trusted, which is the kind of circularity a description rather than an explanation has to live with: the reading is safest when the wait it is reading is short. Two ways to die, three things to count met the same shape from the other side, when the waiting time a plant’s totals named depended on which of its points were fragile.

Buds are not apices #

The no-history start is the one to notice. It names the wait for 3.0 per cent of plants after two seasons lost, 9.6 after four and none at all after thirty. The more of the plant it sees, the worse it does.

Treating every living point as a mature apex amounts to saying that every bud on the plant will branch next season, when in fact the buds branch over the coming wait. The prediction then runs ahead of the plant by the whole stock of waiting buds, and a larger plant makes that error more certain rather than less. The deviance finds the candidate wait whose recurrence least overshoots, which is not the true one. What the window’s start must supply, then, is not the whole lost history but one fact from it: that some of the living points are buds and have yet to branch. The living total split, which knows that and nothing else, reaches 98.2 per cent after ten seasons lost and 100 after twenty. The recurrence a count that loses its growing points wrote down holds the delay in its waiting buds, and a start that empties them has removed the delay from the reading before a season is read.

One share a season, read late #

A late window is kinder to every reading, and the obvious question is whether sorting still earns its place once the plant is large. The control is one dated share a season read from the same window, with the living total split as steady growth would hold it.

In a calm climate one share a season catches up: at a three-season wait it names the wait for 86.4 per cent of plants after ten seasons lost and 99.8 after twenty, against the sorted reading’s 98.7 and 99.9. Under bad years it never does. After thirty seasons lost it names a three-season wait for 85.3 per cent of drought plants and 81.2 of frost plants, while the sorted series names it for every one.

The difference is a wrong model made precise. One share a season assumes that a bud and an apex die at the same chance, and they do not — 0.15 against 0.05 on average. In a calm climate the mix of buds and apices on a steadily growing plant is fixed, so a single chance averaged over the mix is wrong by a constant, and the constant is absorbed. A bad year does not average out: after a bad year the mix shifts, the averaged chance is wrong by a different amount each season, and on a plant of a thousand points the deviance sees the error as clearly as it sees the wait. More data does not rescue a model that is wrong; it makes the wrongness decisive. At a two-season wait the effect is small — one share a season names it for 99.9 per cent of drought plants after twenty seasons lost — because a shorter wait holds fewer buds and the mix moves less.

How short a late window can be #

A large plant also needs fewer seasons read.

Read from the first season, three seasons name a three-season wait for no plant at all, since no bud has had time to mature, and four name it for 82.6 per cent. After twenty seasons lost, three seasons read name it for 88.5 per cent and four for 96.0. The window no longer has to contain a whole wait, because the buds maturing inside it were made before it began, and the steady-growth start has already said how many of each age there are. After two seasons lost the window is worst of all — 13.9 per cent over three seasons read and 37.0 over four — because both of the start’s failures meet there: a plant too small to be informative and too young to be steady.

What a late window cannot read better #

Everything so far improves with the plant’s size. One thing does not, and it marks the limit of what a late window buys.

The averaged chances are read as the dated reading reads them: each season’s chance from its own scars, one season one vote, over the seasons holding ten or more points of the kind. From the first season a ten-season window reads the bud chance to 0.14 and the apex chance to 0.06, because most of its seasons hold too few points to vote. After ten seasons lost the bud chance is read to 0.046 in a drought, and after twenty to 0.040 — and there it stops. After thirty seasons lost it is 0.041.

The stop is the climate. A bud’s chance is five times its good-year value one season in ten, and ten seasons hold on average one bad one, sometimes none and sometimes three. The average over ten seasons of a chance that moves like that has a spread of its own, 0.0407, however precisely each season is read, and the late window’s plant reaches that floor at about twenty seasons lost. The drought’s apex chance meets its own floor, 0.0136, at the same place. The frost’s apex chance, which a frost never touches, has no floor: it falls from 0.0132 after ten seasons lost to 0.0034 after twenty and 0.0008 after thirty, improving as fast as the plant grows.

So a late window reads the plant — its wait, its apex chance where the weather leaves it alone — as well as its size allows, and reads the climate only as well as the number of seasons allows. Scars with dates on them put it as the averaged chance coming only as fast as seasons do. Losing early seasons costs the climate reading nothing, because the early seasons were too small to vote, and a larger plant cannot buy it anything either.

The frost test gains #

The last question was the frost test: the dispersion of each season’s bud deaths about the share a common ratio of chances predicts, which is near one when a bad year strikes both kinds alike and large when a frost strikes buds alone. Read from the first season it waited for the plant to grow, flagging about 41 per cent of frost plants over twenty seasons and 91 over forty.

It loses nothing and gains at every loss. Over twenty seasons read it flags 42.1 per cent of frost plants from the first season, 78.0 after ten seasons lost, 87.4 after twenty and 90.0 after thirty; over forty seasons read, 90.8, 97.2, 98.8 and 99.0. The threshold falls as the loss grows, from 1.83 to 1.46 over twenty seasons, because the drought plants’ dispersion tightens toward one as their seasons fill with deaths. The first seasons were exactly the seasons the test could not use: fewer than five deaths, a few buds, a ratio that could not be read. One frost season is seen once the plant carries a few hundred buds, and a late window starts with them.

What this does not establish #

That a field specimen’s scars can be dated and sorted inside the window, or its living points told apart by kind at the window’s start. Both are assumed, as they were from the first season. The lost seasons are lost whole here; a real plant loses its oldest scars gradually, with the first seasons partly legible, and a partly legible season is a reading between the two measured. The climate is stationary — the same chance of a bad year every season — so a late window is as representative of the climate as an early one. A plant whose climate changed over its life would make the lost seasons carry something the window cannot recover.

What would overturn it #

A plant large enough that the steady-growth start misplaces a twentieth of its buds, read over ten seasons, whose wait the sorted series fails to name. A late window on which one share a season, in a climate with bad years, names a three-season wait as often as the sorted series. A frost test that loses power as seasons are lost. Each would mean the reading here is wrong about what the early seasons carried.

Still open: a climate that changes #

Every plant here grows under one climate, so the seasons it has lost looked like the seasons it keeps. A real plant may have started life under a different regime — a run of droughts in its youth, a warmer decade since — and then the window reads the present climate while the plant’s size and age structure were set by the past one. The next measurement grows plants whose chance of a bad year, or its severity, changes partway through their lives, and asks three things of a late window: whether the steady-growth start, which assumes the window’s survivals held all along, misplaces the buds’ ages; whether the wait is still named; and whether the sorted series can see the change itself — a plant whose age structure disagrees with its window’s survivals is a plant whose past was different, and that disagreement is a reading.