Scars with dates on them

A branching plant whose buds wait before they branch has a count that grows at a rate set by the wait, and a count set by a delay showed that a one-season wait is what produces Fibonacci’s numbers. A count that loses its growing points let points die and found the rate simply scaled by the survival, and what a scar is worth found that counting the scars recovers what death takes away: a rate and a scar share, together, name both the death chance and the wait.

A bad year does not average out broke that reading. It let every growing point on a plant share one season’s death chance — bad one season in ten, good otherwise, with the average held at 0.1 — and found the expectation untouched and no plant near it. The scar share turned out to be set by how long ago the last bad year was, and the reading of the wait got worse the longer it ran: right for 57 per cent of plants over ten seasons and 22 per cent over eighty, with bad years killing half the points.

That essay read the scars as one total. It ended by pointing out that on many plants a scar can be dated. A shoot that died leaves its stump at a place along its parent, and the parent’s growth between bud scales, or the ring the stump sits in, says which season it died. A dated scar count is a series — deaths in each season — and this essay asks what that series is worth.

Two readings of one plant #

The totals reading is the one the bad-year essay used. Take the rate the living count grew at over the window, after the first few seasons, and the scars per living point at the end; invert the two closed forms the scar essay found for the death chance and then the wait.

The dated reading uses the series. In each season, the scars made that season divided by the points then living give that season’s death share. The bad years stop being a nuisance to be averaged over and become a known input. For each candidate wait, the branching recurrence is run from one apex through the plant’s own seasons, each with its own survival, and the wait whose predicted counts best follow the plant’s living counts is the one named.

In the idealised plant of the earlier essays, whose every season’s deaths are exactly its chance times its points, the dated reading would be right every time, from any window. So the plants here are finite. Each starts as one apex, and every growing point — each mature point and each bud at each age — lives or dies by its own draw in every season. What can still go wrong is what a count of scars cannot record: whether the points that died that season were mature points or buds, and how small a young plant is.

The plant in the figure lives through two bad seasons while it is large enough to see them, at seasons 11 and 17, and loses about half its points in each. Read as two totals over twenty seasons it names a wait of one season: the last bad year sits three seasons before the end, and its lump of scars inflates the scar share, so the death chance recovered is too high and the deathless rate too fast. Read as a series, the two-season recurrence run through its own seasons follows it step for step — the dips at the bad seasons included — while the one-season and three-season recurrences diverge from the first bad year on. Turn the dial and the same draws are replayed with milder or harsher bad years: the totals name two seasons only when no season is worse than another and one at every other setting, and the series names two every time.

More seasons, read the other way #

One plant is an anecdote. Two thousand plants a point make it a measurement.

With bad years killing half the points, the totals name a two-season wait for 50.9 per cent of plants over ten seasons, 44.9 over twenty, 29.5 over forty and 21.8 over eighty. These are finite plants, and the bad-year essay’s idealised ones gave 57 and 22 per cent at the two ends, so a plant’s own randomness adds little to what the seasons already did. The dated reading names it for 83.6 per cent over ten seasons, 94.4 over twenty, 99.3 over forty and every plant over eighty.

The misses at ten seasons are about size, not seasons. A plant with a two-season wait holds a median of 23 living points after ten seasons and 325 after twenty, and in its first few seasons, with two or three points, one death is a third of the plant and the recurrences for neighbouring waits have not yet separated. The totals fail differently, and worse: like the fifty junctions of a sample that is confidently wrong, they return a definite answer rather than a vague one, and nothing in the answer says it is wrong.

So dating the scars reverses the direction of the evidence. Undated, more seasons read worse, because the scar share is ruled by the last bad year and a longer window only gives the plant more time to have had one recently. Dated, more seasons read better, as a measurement should, because every season adds a step the recurrence has to follow.

The dashed line is the totals with no bad years at all: every season kills a tenth of the points. That reading improves with the window too — 56.8 per cent over ten seasons, 81.0 over twenty — but the dated reading with bad years beats it at every window short of forty seasons. The series is worth more than the two totals even when nothing is wrong with the totals, because it uses every season and not two numbers made from them.

The bad years stop mattering #

The sharper test is how much the badness of a bad year matters to each reading, holding the average fixed.

Over twenty seasons the dated reading names a two-season wait for 95.1 per cent of plants when no season is worse than another, 95.4 when bad years kill three points in ten, 94.4 at half and 94.2 at seven tenths. The totals fall from 81.0 to 64.4, 44.9 and 32.4. The dated reading does not average over the bad years, it conditions on them, and a condition that is known costs nothing.

What its misses do depend on is the plant. At seventy per cent a bad year also kills more plants outright — three in ten die out within twenty seasons against thirteen in a hundred with no bad years — and those are left aside here, as they would be in a field where a dead plant has no count to read.

A longer wait is harder, and the series still wins #

A two-season wait is the delay the earlier essays measured. A longer one grows more slowly, so a plant takes longer to become large enough to read, and its predicted counts for neighbouring waits sit closer together.

With a three-season wait and bad years at half — a median of fifteen living points after ten seasons — the dated reading names it for 63.7 per cent of plants over ten seasons, 82.5 over twenty, 94.3 over forty and 99.6 over eighty. The totals give 37.6, 37.8, 21.8 and 15.6 per cent. Over ten seasons neither reading has much to go on: a plant with a three-season wait holds about a dozen points by then, and the recurrences for waits of two, three and four have barely separated. The dated reading’s advantage is that it keeps improving; the totals’ is that there is nothing to improve.

A one-season wait, the Fibonacci case, is the easiest of all. The dated reading names it for 97.9 per cent of plants over ten seasons and every plant from twenty, while the totals give 66.9 per cent over ten seasons and 40.4 over eighty. The question a count set by a delay posed — whether a branching count is Fibonacci because its buds wait one season — is answerable from a dated scar series under bad years, and not from a rate and a scar share.

The climate is not the plant’s to read #

The bad-year essay also asked whether the averaged death chance could be recovered, the number the independent model had treated as a property of the plant. A dated series gives each season’s chance directly, so the averaged chance is their mean.

Weighting each season equally, the error falls from 0.049 over ten seasons to 0.034, 0.022 and 0.015 over eighty. The dashed line is the floor no plant can go below: the number of bad seasons in a window of $T$ is itself a draw, and the averaged chance of the seasons a plant lived through differs from the climate’s by $(q_\text{bad} - q_\text{good})\sqrt{\pi(1 - \pi)/T}$ — 0.0149 over eighty seasons. The plant’s own reading sits within five per cent of that from forty seasons on. It is not the plant that limits the reading; it is how many seasons have happened.

With no bad years, the same equal-weight reading is good to 0.0053 over eighty seasons. Reaching that with bad years at half would take more than six hundred seasons, longer than most trees live. So the averaged chance, which the scar essay read as a trait of the plant, turns out under bad years to be a trait of the climate the plant grew in, and a single plant is a small sample of it.

Weight the seasons, not the points #

The figure’s other line is the obvious alternative: total scars over total point-seasons, which gives each dead point one vote. With bad years at half it does not converge at all. Its error is 0.061 over twenty seasons and still 0.056 over eighty.

The reason is the growth. A plant’s points are nearly all recent — at a rate of 1.3 a season, two thirds of every point-season it has ever lived were lived in its last four seasons — so a per-point average is an average over the last few seasons, and it is ruled by whether one of them was bad, exactly as the undated scar share was. Weighting each season equally is what turns a plant’s whole history into a sample of its climate.

With no bad years the ranking reverses. The per-point reading is good to a few millionths over eighty seasons, because every season’s chance is the same and the late seasons simply hold more points to read it from. The equal-weight reading is held back by its first seasons, when the plant had three or four points and one death was a quarter of them.

So the right way to read a dated series depends on whether its seasons differ. That is not something to assume, because the series can be asked.

A plant can tell it had bad years #

If every season has the same death chance, the per-season shares scatter about their mean only as much as a binomial draw at each season’s size allows: widely when the plant held four points, narrowly when it held four hundred. Bad years add scatter that no size removes. The ratio of the observed scatter to the binomial scatter, over the seasons in which the plant held at least ten points, is near one for a plant of uniform seasons and well above one for a plant that lived through a bad year.

Set the threshold on two thousand plants with no bad years, so that five in a hundred are flagged by chance. Over twenty seasons the ratio then flags 66 per cent of plants whose bad years kill three points in ten, 73 per cent at half and 74 per cent at seven tenths; over forty seasons, 95 to 97 per cent; over eighty, all of them. Its misses at twenty seasons are mostly plants that have not yet lived through a bad season while large enough to read one, and nothing could flag those.

So the choice between the two weightings can be made from the series itself. A plant whose ratio stays near one is read per point, and its averaged chance is as good as its size allows. A plant it flags is read per season, and its averaged chance is as good as its climate allows — and the same flag says the two totals are not to be trusted for this plant at all, which is the kind of warning a measurement ought to carry.

Reading the bad year itself #

The plant’s rate is most sensitive to the bad years’ own chance, and the bad-year essay asked whether that could be read from the same series. It can, from the bad seasons alone, once the plant has lived through one while large enough.

Over ten seasons few plants have: a bad season comes one year in ten, and a young plant with a handful of points cannot distinguish a bad season from a good one that happened to be unlucky. By twenty seasons 61 per cent of plants have seen one while holding twenty points or more, and the bad seasons’ chance is read to 0.060. By eighty seasons every plant has, and it is read to 0.011 — about as well as the averaged chance, and from the same series.

What the reading assumes #

It keeps the branching grammar itself, which describes rather than explains: a rule for how a bud becomes a branch, with no light, water or mechanics in it. Everything measured here is about reading that grammar’s one parameter back from a plant that obeys it.

It assumes a scar can be dated to the season. Along a shoot with clear bud-scale scars, or in wood with clear rings, that is ordinary practice; in a herbaceous plant whose dead shoots rot away it may be impossible, and then the totals are all there is.

It counts every dead point as a scar. A count that has lost tips found that a lost tip can leave no trace at all, and a dated series with gaps in it is a different instrument, whose gaps fall hardest on the seasons that killed most.

It gives a bud and an apex the same chance in a bad season. Two ways to die, three things to count found that separating the two chances needs scars sorted by kind; dated scars sorted by kind would give that essay’s third count season by season, and nothing here says how well.

And it takes the seasons to be independent. Droughts run in spells, which would make the averaged chance slower still to read, since a spell is one draw rather than several. The dated reading of the wait would not notice, for the reason the bad years stopped mattering: it runs through whatever seasons the plant had.

Readings that would undo it #

A window over which the dated reading names a two-season wait less often than it does over a shorter one. A bad-year chance at which the dated reading, over twenty seasons, falls more than two points below its rate with no bad years. An equal-weight reading of the averaged chance that beats the seasons’ own sampling error by more than a few per cent. Any would mean the recurrence is not doing what it is claimed to do.

Still open: scars sorted by kind as well as by date #

The dated series here counts every scar alike. The scar essay’s sibling needed scars sorted into buds and apices to separate two death chances, and failed to without them. A dated series sorted by kind is the natural next instrument: each season’s deaths split into buds and apices, so each season gives two shares rather than one. The measurement is whether that series separates a bud’s chance from an apex’s under bad years that strike both — and whether a bad year that strikes buds harder than apices, as a late frost would, shows up as a season whose two shares disagree.