A count that has lost tips

Reading a tree’s radii against the number of tips each branch carries kept a tree built at Murray’s three apart from one built at Da Vinci’s two at every error that could be quoted, where fitting junction by junction had collapsed them onto each other at twelve per cent. The reason was a single property of the regressor: a count is not measured with error, so the noise on the radii sat on one side of the regression only.

That property belongs to a count of the tips a tree grew. A person standing under a crown counts the tips it has now, and the two differ by every shoot that died back, was browsed or broke off. The question here is what that difference does to the instrument whose whole advantage was that nothing was wrong with its count.

The wood stays and the count goes #

When a shoot dies, the branches beneath it do not lose the wood they laid down while it was alive. Every radius on the path from that shoot to the trunk was sized for a count that included it, and every one of them keeps that radius. What changes is the number a person writes down beside it.

So the error a loss introduces is on the count’s side of the regression, which is exactly the side the count was chosen for keeping clean. A relative error on a radius shifts log r; a lost tip shifts log N. The first adds scatter above and below a straight line whose slope survives; the second moves points sideways along the axis the slope is measured against.

That is a different kind of damage from anything a relative error on the radii or a swelling at the fork does, and it is worth measuring rather than guessing, because the direction of the damage is not obvious from the description.

Two ways to lose a tip #

Two models of loss are drawn, and they are meant to bracket what trees do rather than to describe any one of them.

Tips lost one at a time. Every tip is lost independently with a stated probability, the way shoots die back, are browsed or fail to flush. A branch whose tips are all lost is gone; every other branch keeps its radius and carries fewer tips.

Limbs shed whole. Every branch that is neither a tip nor the trunk is shed with a stated probability, taking everything above it, the way a limb snaps in a storm. One event removes a whole run of tips at once.

Both are drawn on the same fifty-junction tree the count was first measured on, 101 segments and 51 tips, three hundred replicates at each level of loss. The losses come from their own seeds, the measurement noise from the earlier seeds, and a replicate applies one draw of losses to both trees, so the tree at three and the tree at two lose exactly the same branches. With nothing lost, every row reproduces the earlier one replicate for replicate: 2.996 [2.81, 3.19] and 1.997 [1.91, 2.08] at twelve per cent.

One replicate with half its tips at risk #

Take one replicate in which each tip had an even chance of being lost. Twenty-three of the 51 tips went, and 72 of the 101 segments survive. Read against the tips still there, the tree built at three gives an exponent of 2.800. Read against every tip it grew, on the same measured radii, it gives 2.985.

The tree built at two gives 1.874 and 1.993 on the same replicate. And the junction fit, over the 27 junctions whose two daughters both survive, gives 2.332 and 1.842: nearer each other than the two counts are, and far below the tree at three’s truth.

Why the branches move and the twigs do not #

The picture shows where the displacement comes from, and the arithmetic is short. A tip that survives is still counted at one, so it sits exactly where it sat before: at the left edge of the plot, log N = 0. A branch carrying forty tips that loses about half of them is counted near twenty, and moves left by about log 2 = 0.69.

The tree’s counts span log 51 = 3.93 in total. Pulling the right-hand end of that span in while the left-hand end stays put compresses the regressor, and the radii, which did not change, now rise over a shorter horizontal distance. The slope steepens, and an exponent is the reciprocal of the slope, so a count that has lost tips reads the exponent low.

The same arithmetic says why the damage grows faster than the loss. A branch that loses a share q of its tips moves by log(1/(1 − q)), which is 0.11 at a tenth, 0.69 at a half and 2.30 at nine tenths — while the surviving twigs never move at all.

One factor for both rules #

With no measurement error at all, the displacement can be written down exactly. Every radius satisfies $\log r = \tfrac{1}{p}\log N_{\mathrm{grown}}$, so the regression of $\log r$ on $\log N_{\mathrm{now}}$ has slope $\operatorname{Cov}(\log N_{\mathrm{now}}, \log N_{\mathrm{grown}})$ divided by $p\operatorname{Var}(\log N_{\mathrm{now}})$, and the exponent it reads is p times κ, where

$$\kappa = \frac{\operatorname{Var}(\log N_{\mathrm{now}})}{\operatorname{Cov}(\log N_{\mathrm{now}}, \log N_{\mathrm{grown}})}$$

— a number computed from the tree and its losses and containing no p anywhere. In one draw that took 13.7 per cent of the tips, κ is 0.9806, and the two trees read 2.942 and 1.961. In one that took 45.1 per cent it is 0.9448, reading 2.834 and 1.890. In one that took 74.5 per cent it is 0.776, reading 2.328 and 1.552.

Wrong about the exponent, right about which is steeper #

The ratio of the two readings is three halves to the last digit in every draw, and that is the useful half of the result. A loss does not blur the difference between the two rules; it shrinks both readings towards zero by the same proportion. A count that has lost tips is wrong about the exponent and still right about which of two trees conserves more.

Averaged over draws, with tips lost one at a time, the factor is 0.983 at a tenth of the tips gone, 0.944 at three tenths, 0.886 at a half, 0.789 at seven tenths and 0.549 at nine tenths. The tree at three reads 2.66 with half its tips lost, which is a number a person could take for a tree between the two rules. It is not. It is Murray’s tree, counted after a season of dieback.

How long the two trees stay apart #

Add back twelve per cent of error on every radius. With a tenth of the tips lost, the count’s two intervals clear each other by 0.70, barely less than with none. At three tenths the tree at three reads 2.825 [2.60, 3.06] and the tree at two 1.884 [1.77, 1.99], still 0.61 apart. At a half, 2.652 [2.39, 2.93] and 1.769 [1.62, 1.92], apart by 0.47.

At seven tenths of the tips gone, 2.366 [2.01, 2.71] and 1.576 [1.36, 1.77], still apart by 0.23. At eight tenths they overlap by two hundredths. The count keeps the two rules apart until a tree has lost about three quarters of what it grew, with every reading of the exponent itself wrong by then by a fifth or more.

The junction fit gives out first #

The junction fit on the same radii was already at its limit before anything was lost: at twelve per cent its two intervals clear each other by a hair, as the count’s first measurement recorded. A loss does not displace it — with no measurement error it reads 3.000 and 2.000 under any loss, because every surviving junction still obeys its rule — but it thins the sample. Half the tips lost leaves 24.6 junctions of the fifty on average, and seven tenths leaves 14.3.

Fewer junctions means a wider interval at the same displacement, and the fit had no margin to spend. With a tenth of the tips lost it still clears by half a hundredth. With a fifth it overlaps. On the same trees, then, the count overlaps only once a tree has lost four times the share of its tips that the junction fit can stand: 79.7 per cent against 19.9.

Leaving out the twigs #

The obvious repair is to leave out the branches where a lost tip is a large share of the count, and to fit only branches still carrying four tips or more. It does remove part of the displacement: with three tenths of the tips gone the tree at three reads 2.949 instead of 2.825.

It pays for that in precision, and the price is larger than the purchase. Its interval at three tenths is [2.33, 3.73], three times as wide as the full count’s, and the two trees’ intervals clear by only 0.02. At four tenths they overlap, where the full count is still 0.55 apart. The twigs the repair discards are the same points that give the regression its spread, which is the property the count’s information came from in the first place.

Counting what died #

The repair that works is to count what the tree lost as well as what it has. A shoot that dies leaves a scar on the branch that bore it, and a count of scars plus shoots is a count of every tip the branch grew.

Read that way, the tree at three returns 3.000 [2.80, 3.25] with half its tips gone and 2.998 [2.73, 3.33] with nine tenths gone, and its interval stays 0.58 clear of the tree at two’s at the heaviest loss drawn. With no measurement error it reads three exactly under any loss. The only thing a scar count gives up is the handful of branches that vanished altogether, which costs a little spread and no displacement.

What it asks of a person is care rather than instruments: scars are small, a thickening branch can grow over them, and a scar that has been overgrown is a lost tip again.

A limb takes a run of tips #

Shedding whole limbs is a different experiment, and it is harsher at every share of tips lost. A two per cent chance of shedding each internal branch removes 11.5 per cent of the tips on average; five per cent removes 27.9; ten per cent removes 48.1.

At 11.5 per cent the count’s intervals clear by 0.62, not far from single losses. At 27.9 per cent they clear by 0.15, against 0.61 for a similar share lost one at a time. At 48.1 per cent they overlap by a third of a unit, where single losses at the same share were still apart by 0.47. On the fifty-junction tree the count survives about a quarter of its tips lost in limbs, against seven tenths lost singly.

Why a limb is worse than its tips #

The means say something unexpected. At 48.1 per cent of tips lost in limbs the tree at three reads 2.593, and at 49.9 per cent lost singly it reads 2.652. A limb does not pull the reading much further down than the same number of tips taken one at a time.

What it does is spread the reading out. The interval is [1.69, 3.05] against [2.39, 2.93] — two and a half times as wide. Tips lost singly take close to the same share from every large branch, so every branch moves by about the same amount and the factor κ hardly varies from draw to draw. A limb takes nothing from most branches and a large share from the few beneath it, and whether the trunk’s own count loses a fifth or a half depends on which limb broke. The factor becomes a lottery, and a lottery is what an interval measures.

A stub is not a scar #

A shed limb does leave something behind, a stub where it joined its parent. So the natural extension of counting scars is to count each stub once, beside the tips that remain. Under single losses that is the scar count exactly, row for row, because every lost tip is its own stub.

Under shed limbs it makes things slightly worse. At 11.5 per cent lost the stub count’s intervals clear by 0.52 against the plain count’s 0.62; at 27.9 per cent, by 0.02 against 0.15. The reason is the same arithmetic as before: a stub adds one to every count beneath it, whether the limb carried two tips or twenty, so it moves branches by amounts unrelated to the wood that was lost.

The reading that would repair a limb is a count of every tip it carried, and that does keep the trees apart to 86.4 per cent of tips lost. But nothing on the tree records it. A shed limb has no countable repair; its loss has to be reported as a wider interval.

On a larger tree #

Two hundred junctions give every instrument more to work with, and the ordering does not change. With tips lost one at a time the count keeps the trees apart at every loss drawn, to 89.9 per cent, clearing by 0.24 there; the junction fit is apart to 69.7 per cent and overlaps by 79.8.

With limbs shed, the count is apart to 61.7 per cent of tips lost and overlaps by 76.2; the junction fit is apart to 37.9 and overlaps by 61.7. The count outlasts the junction fit under both kinds of loss, and both instruments outlast limbs worse than tips, on the larger tree as on the smaller one. A larger sample buys margin without changing which damage is worse.

At twenty per cent of error #

The thresholds belong to the loss rather than to the error on the radii. At twenty per cent of error the fifty-junction count is still apart to seventy per cent of tips lost singly and overlaps by eighty — the same levels as at twelve — with a thinner margin of 0.08 at the last. Shed limbs still cost it the separation between 27.9 and 48.1 per cent.

The junction fit at twenty per cent overlaps with nothing lost at all, reading 1.292 and 1.254, the inversion already under way. A loss has nothing left to take from it. The count at the same error reads 2.998 and 1.997 before any loss, and 2.657 and 1.769 with half its tips gone.

What this changes for a person with a crown to count #

Three pieces of practice follow, each a consequence of a measured number rather than a recommendation.

Count scars as well as shoots. Under single losses that removes the displacement entirely, and a scar count at twelve per cent of error keeps the two rules apart at every loss drawn.

Do not read an exponent between the rules as evidence of an intermediate rule on a tree with visible dieback. A tree at three that has lost half its tips reads 2.66, and with just over a quarter of its tips lost in limbs its interval already reaches below 2.2.

Where limbs have been lost, report the interval, not a corrected reading. No count a person can make undoes a shed limb, and a stub counted once widens rather than narrows the answer.

Losses nobody drew #

Both models lose tips without regard to where they are or how large a branch they hang from, and real losses are rarely that fair. A crown that sheds its shaded inner shoots loses by position; one that loses its weakest twigs loses by size; browsing takes the shoots an animal can reach. Each of those correlates the loss with the count, which neither model does, and nothing here says which way that pushes the reading.

The tree is one Yule draw of fifty junctions and one of two hundred. The error on the radii is the same relative Gaussian error the duel used, which makes a radius non-positive often enough past thirty per cent that rows there are not quoted. And a scar is assumed countable: on a trunk that has self-pruned for decades, the early scars are buried in wood.

What would withdraw it #

With no measurement error, a count reading whose ratio to its rule differs from κ in any draw. A scar count or a junction fit, with no measurement error, reading anything but the built exponent under a loss. On fifty junctions at twelve per cent, the count overlapping at or before the loss at which the junction fit overlaps, a scar count overlapping at any single loss, or shed limbs costing the separation at no smaller a share of lost tips than single losses. Each of these is checked every time the measurement runs.

Still open: a loss that chooses its branches #

The next test takes away the fairness of both models: tips lost preferentially from the smallest branches, or from the inner ones, so that the chance of a loss depends on the count it changes. The questions are whether such a loss still lowers both rules by one factor, which direction it moves the reading, and whether counting scars still restores it — since a sample chosen by what is reachable has already turned out to be the dangerous one for the junction fit.