A frost and a drought in the scars

A branching plant can be read from its counts — a description, not an explanation, but a description with numbers in it — and the reading has been refined one assumption at a time. A count set by a delay found that the wait before a new shoot branches fixes the rate the living points multiply at; what a scar is worth added death, one chance for every growing point, and found that a rate and a scar share name the chance and then the wait. Two ways to die, three things to count gave waiting buds and mature apices chances of their own, and found the two separable from a rate and a scar share only badly — their loci cross at eight to sixteen degrees — so that a third count was owed: the scars sorted by kind.

Then the weather arrived. A bad year does not average out let one season in ten be bad for every point at once, and found that no plant’s totals sit near their expectation, because the scar share is set by how long ago the last bad year was. Scars with dates on them repaired that by reading each season’s death chance off its own scars and running the branching recurrence through the plant’s own seasons, which named a two-season wait for 94 per cent of plants over twenty seasons whatever the bad years killed.

The two repairs were made in different plants. One had two chances and no weather; the other had weather and one chance. A real plant has both, and bad years need not strike buds and apices alike — a late frost kills buds and spares mature shoots, a drought presses on everything. This essay sorts the dated scars by kind and asks what the combination can read.

Two chances and two kinds of bad year #

The plants are finite and grown point by point from one apex, as the dated plants were: mature apices and buds of each age, every point dying by its own draw each season. Averaged over the climate an apex dies at 0.05 a season and a bud at 0.15 — the fragile-bud plant the two-chance essay worked — and the wait before a new shoot branches is two seasons unless stated. One season in ten is bad, in one of two ways. In a drought both kinds’ chances are $k$ times their good-year values, so the bud chance is three times the apex chance in every season. In a frost only the buds’ chance is $k$ times its good-year value and the apices die at 0.05 every season. Good years are set so that the averages hold, and $k = 1$ is a calm climate.

At $k = 5$ a drought takes a good year’s 0.036 and 0.107 to a bad year’s 0.179 and 0.536; a frost takes the buds from 0.107 to 0.536 and leaves the apices alone.

The picture is one plant through twenty seasons of frost at five times a good year. It meets bad seasons at 6, 10, 11 and 18. In season 18 it holds 135 apices and 163 buds, and loses 6 apices and 98 buds: an apex share of 0.044 and a bud share of 0.60. Nothing in a single total says that the second of those is what happened; the sorted series says it at a glance. Early on the plant is too small for either share to mean much — in season 6 it holds six apices and seven buds.

Sorting names the wait sooner #

The reading is the dated reading with one change. Each season’s apex chance is read as its apex scars over the apices then living, its bud chance likewise, and the recurrence is run from one apex through both survivals for each candidate wait; the wait whose predicted living counts best match the plant’s is named.

Over ten seasons the sorted and dated series names the wait for 97.3 per cent of frost plants, where one dated share a season names it for 81.2. By twenty seasons both are near every plant, 99.8 and 97.2, and from forty they agree at all of them. The same gap stands at ten seasons in a drought, 97.1 against 81.5, and in a calm climate, 97.7 against 82.4. So the bad years are not what sorting repairs in the naming of the wait; the dated series had already repaired that. What sorting adds is sooner.

Why sorting helps even when both kinds die alike #

The obvious explanation is that one chance a season is the wrong model when buds and apices die at different rates. It is part of the answer and not the most of it.

Give the apices and the buds the same chance, 0.1, and the one-share reading is no longer wrong about anything. It still names the wait for 82.9 per cent of ten-season plants against the sorted series’ 97.1. The reason is in the plant’s first seasons, when it holds a handful of points. Whether a season’s two deaths fell on two apices or two buds is chance, and it matters a great deal to how the plant grows next: an apex lost is a branch that will not branch, a bud lost is a branch that never arrives. One share a season averages over that luck. The sorted series runs the recurrence through the survivals that actually happened to each kind, so its predicted counts follow the plant’s own early accidents rather than their average.

The wrong model adds to that when it is wrong. With apices dying at 0.15 and buds at 0.05 — the fragile-apex plant — one share a season names the wait for 69.0 per cent of ten-season plants against 97.9. And a longer wait widens the gap: in a frost with a three-season wait, 51.5 against 90.7 over ten seasons, and 65.2 against 96.8 over twenty.

The totals the two chances were first owed #

The two-chance essay asked for three totals: the rate the living points multiply at, the scars per living point, and the apices’ share of all scars. In expectation, and without weather, those three pin both chances to about a hundredth and name the wait. They are what a plant offers when its scars can be sorted but not dated, and it is worth knowing what they are worth once the seasons differ.

In a calm climate the three totals work, slowly: they name the wait for 37 per cent of plants over ten seasons, 74 over twenty, 99.5 over forty and every plant over eighty, and read the bud chance to 0.0086 over forty seasons and 0.0040 over eighty. With bad years at five times a good one they do not settle at all. Over eighty seasons they name the wait for 51.7 per cent of drought plants and 73.3 per cent of frost plants, and the bud chance is out by 0.127 and 0.114 — most of the chance itself — at twenty seasons as at eighty.

That is the bad year’s lesson again, now for the split. The apices’ share of all scars, like the scar share before it, is dominated by the last bad season, because a growing plant’s points are nearly all recent. Sorting the scars without dating them hands the reading a third number with the same fault as the second. The dated and sorted series, weighting each season equally, reads the bud chance to 0.0159 in a drought and 0.0160 in a frost over eighty seasons.

Each kind’s chance, read #

What each kind’s chance can be read to is set by the climate rather than the plant, as it was for the single chance.

Over eighty seasons a calm plant’s chances are read to 0.0019 for the apices and 0.0031 for the buds. In a drought the apex chance is read to 0.0056 and the bud chance to 0.0159; in a frost the bud chance to 0.0160 and the apex chance to 0.0018. The bud chance in both bad climates sits within fifteen per cent of the error the seasons themselves impose, $(q_{\text{bad}} - q_{\text{good}})\sqrt{\pi(1 - \pi)/T}$, which is 0.0144 here; the dated essay found the same bound for the single chance, and no plant can beat it, because the climate’s average is only as well known as the number of seasons sampled.

The frost’s apex chance is the telling number. A frost does not touch the apices, and the sorted series reads their chance as if there had been no bad years at all — 0.0018 against a calm plant’s 0.0019. A dated series of one share a season could not have done that; it would have read one averaged chance, struck by every frost, and left the split to the expectation arithmetic that the weather breaks.

The ratio of the two chances tells the same story from the other side. The plants’ buds die three times as readily as their apices on average in every climate here. In a drought that ratio holds in every season, good and bad, and eighty seasons read it to within about four per cent. In a frost it holds in the good seasons and jumps fivefold in the bad ones; the ratio of the two averaged chances is still three, but it is an average over two different ratios, and eighty seasons read it only to about twelve per cent. A single number for how much more fragile a bud is than an apex is a fair summary of a drought plant and a poor one of a frost plant, and the sorted series is the only record that shows which kind of plant it is summarising.

Which kind of bad year #

The sorted series can ask something no other reading can: whether its bad seasons struck both kinds alike. If the two chances keep one ratio every season, then given how many points died in a season, the share of them that were buds is fixed by how many buds and apices were living. In a drought that holds season by season. In a frost it fails in exactly the bad seasons.

So the test fits the one ratio that makes the plant’s bud deaths sum to what it predicts, and measures how far each season’s bud deaths stray from that prediction, against the spread chance alone allows. Near one, the seasons are consistent with a single ratio; large, they are not. The threshold is set on drought plants of the same severity so that one in twenty is flagged.

Over twenty seasons it flags 41.0 per cent of frost plants at five times a good year and 25.3 at three; over forty, 90.7 and 86.8; over eighty, 99.6 at both. Over ten seasons it cannot be read at all, because almost no plant has three seasons of five deaths.

Against the same threshold, a calm plant — no bad years at all — is flagged 4.6 to 5.9 per cent of the time, as often as a drought plant. That is the test doing what it was built for and nothing else. A drought and a calm climate both keep the bud chance three times the apex chance in every season, so they are the same thing to a test about the ratio; what separates them is whether the seasons’ death shares vary at all, which is the dated essay’s dispersion of the single share. The two tests answer two questions — were there bad years, and were they frosts — and a sorted, dated record can ask both of the same scars.

The ratio the test fits has to be the right one, and the first version was not. Pooling each kind’s deaths over its own living points weights the seasons differently for the two kinds, and after a bad year a plant’s share of buds shifts, so the pooled ratio is slightly off. On a plant of eighty seasons, which holds tens of millions of points, slightly off is enough: the dispersion read that bias as a frost on nearly every drought plant, and the threshold set on drought plants came out at 819 rather than 1.2 with bad years three times a good one, and at 6,503 with five. The ratio at which the bud deaths sum to their prediction has no such bias, and it is the one read here.

One frost at a time #

The plant-level test says whether a plant’s history held frosts. A single season can be tested too — its bud deaths against the common ratio the plant’s other seasons fit — and the question is how large the plant must be before one frost shows.

A frost at five times a good year is seen in 0.2 per cent of seasons with fewer than ten buds living, 16.6 per cent with tens of buds, 89.7 per cent with hundreds and every one from a thousand. At three times a good year, 1.6 per cent with tens and 69.0 with hundreds. So a young plant’s frosts are invisible in its scars one at a time, and on the ten-season plants that the sorted series names so well almost all are, which is why the plant-level test needs forty seasons to find most frost plants: it waits for the frosts that happen after the plant has grown.

What this changes about reading a plant #

The dated essay’s conclusion was that a plant reads its own wait well and its climate’s average only as fast as seasons come. Sorting keeps both halves and adds three things. The wait is named from a shorter record, because the recurrence follows the accidents of a small plant’s first seasons. Each kind’s chance is read separately, each to the bound its own weather sets — and where a kind is untouched by the weather, as the apices are by a frost, to the calm-climate precision. And the record says what its weather was: a drought and a frost with the same effect on the plant’s total deaths leave different signatures in its sorted scars, and forty seasons tell them apart nine times in ten.

What sorting does not do is rescue the totals. Three undated numbers are no better against the weather than two were, and the third count the two-chance essay asked for turns out to need the date as much as the first two did.

What these plants assume #

That a bud scar and an apex scar can be told apart and dated to the season; both are assumed, and on a real plant the first is a matter of anatomy and the second of growth rings or position along a shoot. The bad years are independent from season to season, and each is one clean mechanism; a real climate mixes frosts and droughts, and a season that is both would read as a drought with a stronger effect on buds, which the test would flag. The plants are finite and start from one apex, and a plant read after it has grown large has lost its early seasons’ scars to wood and bark — exactly the seasons in which sorting helped most.

Readings that would undo it #

A calm plant whose chances are equal for which one share a season names the wait as often as the sorted series over ten seasons. Undated sorted totals that read the bud chance better with more seasons under a bad-year climate. A threshold set on drought plants that flags frost plants no more often than drought plants at forty seasons. A frost season with a thousand buds living that the single-season test misses. Each would mean the reading here is wrong about what sorting adds, rather than incomplete.

Still open: the early seasons a plant no longer shows #

Everything here reads the plant from its first season. A plant found in the field has grown over its oldest scars, and the seasons it keeps are its recent, large ones — the seasons where frosts are visible and the recurrence least needs help. The next measurement is the sorted, dated series read from a window that starts late: how many early seasons can be lost before the wait is named no better than by one share a season, whether the living counts at the window’s start can stand in for the lost history as an initial condition, and whether the frost test, which already waits for the plant to grow, loses anything at all.