A loss that picks its tips can read high

A count that has lost tips found that a tree whose shoots die back reads its branching exponent low. Every branch keeps the wood it grew for the tips it once fed, while the number written beside it falls, and the regression of radius on count steepens. The arithmetic gave the damage a name: the exponent read is the built exponent times a factor κ, the variance of the log of the tips still there over its covariance with the log of the tips grown. A tree built at Murray’s three and one built at Da Vinci’s two share that factor, so a loss shrinks both readings by one proportion and leaves their ratio at three halves.

Both of the losses measured there were fair. Every tip had the same chance of dying, or every limb the same chance of breaking. Real losses are not fair: a crown sheds its shaded inner shoots, starves the weakest twigs, loses the ones an animal can reach. The question that essay left was what a loss does when it chooses — whether it still lowers both rules by one factor, which way it moves the reading, and whether counting scars still restores it.

The factor survives any choice #

The first part has an answer before anything is grown. With no error on any radius, every branch that survives has radius exactly $N_{\mathrm{grown}}^{1/p}$, so the logarithm of its radius is the log of the tips it grew divided by p. The regression is run against the log of the tips it has now, and its slope is the covariance of those two logs divided by p times the variance of the second. Nothing in that expression knows how the tips were chosen.

So the exponent read is still $p\kappa$ with κ computed from the tree and its losses alone, and two trees losing the same tips still read in the ratio three to two whatever rule picked them. Checked on the extreme choices found below, the departure from that identity is below a billionth. A choosing loss cannot blur the difference between the rules any more than a fair one could, which is the property that made the count worth having over the junction-by-junction fit. What it can do is move κ, and the question becomes how far, and in which direction.

The fair loss reads low every time #

On the fifty-junction Yule tree the earlier measurements used — 101 segments, 51 tips — a fair loss of a tenth of the tips gives a mean factor of 0.983, of three tenths 0.944, of half 0.886, of seven tenths 0.789. Every draw is below one, and the reason is that a fair loss takes a steady share from every large branch while the surviving twigs, counted at one, never move. The right-hand end of the count’s axis is pulled in and the left-hand end stays where it was.

That reasoning has a hole in it, and the hole is the whole of this essay. It assumes the large branches lose about the same share. A loss that chose its tips could take most of a few branches and almost nothing from the rest, and nothing in the identity above says κ has to stay below one when it does.

Searching over which tips die #

To find out how far a choice can move the factor, the measurement fixes a share of the tips and asks which tips, making up exactly that share, put the highest factor on the count and which the lowest. There are about $2.5\times 10^{14}$ ways to choose 26 tips of 51, too many to list, so a search does it: start from a random choice, swap one lost tip for one kept tip, keep the swap if the factor moves the wanted way, and repeat four thousand times from each of twenty seeded starts.

A search can stop short of the true extreme. The range it finds is therefore a lower bound on the range a choice can reach: the true highest factor is at least as high as the one found, and the true lowest at least as low. Everything said below about how wide the range is holds with that direction attached.

A factor above one #

With a tenth of the tips lost the choice already matters both ways: the factor found runs from 0.954 to 1.040. At three tenths it runs from 0.863 to 1.093, and at half the tips from 0.743 to 1.173. A fair loss of half the tips sits at 0.886, inside the range and nearer its bottom. The top of the range is above one from the smallest loss measured to seven tenths, where it reaches 1.176, and only at nine tenths, with five tips left on the tree, does it fall back below one.

So a loss can read the exponent high. The same 26 tips, chosen one way, read Murray’s tree at 2.23; chosen another, at 3.52. The tree at two reads 1.49 or 2.35 on the same choices. The earlier conclusion that a count which has lost tips reads both rules low belongs to the fair loss, and the reason it held there is a property of fairness rather than of loss.

What the high choice looks like #

The picture of the high choice is unlike anything a fair loss draws. Of the 64 branches that survive, fifteen carried two tips or more and now carry exactly one. They are branches whose wood was laid down for two, five, nine tips, sitting at the left edge of the count’s axis with radii well above a twig’s. A column of them stands over the count of one.

The largest branches, on the right, have moved only a little. The trunk loses half its tips, as it must at any choice of half, but the next branches below the trunk lose far less than half each, because the losses have been concentrated elsewhere. The cloud has been stretched rather than compressed — the opposite of what a relative error on the radii does, which only scatters it: its left end has gained a stack of heavy points and its right end has barely moved. The fitted line through it is flatter than the true one, and a flatter line reads a larger exponent.

What the low choice looks like #

The low choice is the opposite shape. No branch that grew two tips or more has been reduced to one, so the column at the left is gone; the small branches that survive keep most of what they had. Every large branch, meanwhile, has lost about the same large share. The right end of the cloud has moved left as a block, the left end has not moved, and the line through the cloud steepens past what any fair loss of the same size gives.

Read this way, a fair loss is a compromise between the two: it moves every large branch by about the same amount, like the low choice, but it also strips some small branches to one tip, like the high choice, and the two effects pull κ in opposite directions. The steady compression wins on average, which is why the fair loss always reads low, and only by the margin between them.

Where each size of branch moves #

The same contrast in one picture. A fair loss of half the tips moves branches of four tips or more by about 0.76 on the log scale, the same in every band, and branches of two or three by 0.54. The low choice moves the two largest bands by 0.94 and 0.85, further than the fair loss, and the smallest by 0.14, much less. The high choice moves the middle band, nine to twenty tips, by 1.53, and the largest band, twenty-one or more, by only 0.41.

That is the rule in words. A loss reads low when the displacement it gives a branch rises with the branch’s size and varies little among branches of one size. It reads high when the displacement is large and uneven in the middle of the tree and small at the top. Written as statistics, κ exceeds one exactly when the variance of the displacement exceeds its covariance with the log of the tips grown. It is a systematic error in the sense a swelling at the fork was one — it moves every reading on a tree the same way — but unlike a swelling it has no fixed sign.

The reading that names no rule #

The ratio survives, but a person standing under one tree has no second tree to divide by. They read one number and ask which rule it belongs to, and for that question the range is what matters. With no error on any radius, the tree at three can read anywhere from three times the lowest factor to three times the highest, and the tree at two from twice the lowest to twice the highest.

On fifty junctions the two ranges stay apart through four tenths of the tips lost: there the tree at three reads at least 2.44 and the tree at two at most 2.24. At half the tips they meet — Murray’s tree can read as low as 2.23 and Da Vinci’s as high as 2.35 — and past that point a reading in the overlap could have come from either rule under some choice of losses. On two hundred junctions, whose larger crown gives the search more room at the extremes but also gives the trunk more branches to average over, the ranges first meet between half and six tenths — a larger sample buys margin here as it did for every other damage measured.

That is the sharpest consequence of choice. A fair loss of seven tenths of the tips leaves Murray’s tree reading 2.37 and Da Vinci’s 1.58, nowhere near each other. A chosen loss of half the tips can leave them indistinguishable from one reading each.

Four rules a crown might follow #

The extremes are what an adversary could do with a tree. A crown is not an adversary, and the useful question is where the rules it plausibly follows put the factor. Four are drawn, each losing tips with a probability set by the tip’s rank on one property: twice the nominal rate for the first-ranked tip, nothing for the last, linear between, so the expected share lost stays near the nominal one.

Every tip alike, the fair loss of the earlier measurement. The smallest branches first, ranked by how many tips the tip’s parent carries — weak shoots on thin wood. Nearest the trunk first, ranked by depth — a crown shedding its shaded interior. Furthest out first, the reverse — exposed shoots browsed or pruned by wind. A probability is capped at 0.98, so at the heaviest levels the share actually lost falls short of the nominal one, and everything is drawn against the share actually lost.

All four read low #

None of the four reaches one at any level. With half the tips gone a fair loss reads 0.886, taking the smallest branches’ tips first 0.879, the outermost first 0.872, and the innermost first 0.941 — the least low of the four. At the heaviest levels the order holds: the inner rule reads 0.824 with 68 per cent of the tips gone, where a fair loss of a similar share reads about 0.79.

The rule that pushes hardest toward the high side is the inner one, and it is easy to see why from the profile. A tip near the trunk sits on a short path, so losing it subtracts from the trunk and two or three large branches and from nothing else; the middle of the tree is spared. That is half of the high choice’s shape — the largest branches displaced — without the other half, the middle stripped bare. A rule that ranks tips by one property cannot concentrate its losses in a few limbs, and concentration is what reading high needs.

On two hundred junctions the same order holds and the inner rule reads higher still, 0.950 at half the tips and 0.899 at 68 per cent. A larger tree has more inner tips on short paths, and losing them leaves more of the crown untouched.

One limb dying back #

The other natural shape is concentration itself: one limb in shade, its shoots dying back while the rest of the crown thrives. That is the high choice’s mechanism in miniature, and it is drawn by letting every tip above one chosen branch die with probability 0.8 and every other tip live, for each of the tree’s 32 branches carrying three tips or more in turn.

Five of the 32 read above one, and the highest, a limb of seven tips one segment from the trunk, reads 1.0165 while costing the tree 11 per cent of its tips. The others cluster just under one. A limb dying back is the right shape but the wrong size: it strips one branch’s interior while every branch beneath it, down to the trunk, loses a share of its own count too, and those shared losses are the compression a fair loss makes. The high choice found by search strips four or five middle-sized branches at once and spares their parents.

At a dieback of one half rather than eight tenths the highest limb reads 1.011; on two hundred junctions, 33 of 129 limbs read above one and the highest 1.0098. A crown that sheds one side reads its exponent high, if at all, by a hundredth or two.

The two trees at twelve per cent of error #

Put back twelve per cent of relative error on every radius, the level at which the junction fit had already collapsed and the count had not. The question for a pair of trees losing the same tips is whether the four rules cost the separation sooner than a fair loss did.

They do not. A fair loss keeps the trees apart by 0.47 at half the tips and 0.23 at seven tenths, and overlaps at eight tenths; the inner rule keeps them 0.52 apart at half and 0.37 apart with 68 per cent gone. The other two rules sit within six hundredths of the fair loss at every share, on either side of it. Since the ratio cannot change, a rule that reads less low also widens the gap between the readings, and the inner rule’s higher factor is a slightly larger margin. None of the four comes near the inversion the junction fit reaches at higher error.

Scars, again #

A count of scars is a count of every tip each branch ever carried, and with single losses that is the count grown exactly, whichever tips died. With no error it reads three and two to the last digit under all four rules at every level. At twelve per cent of error and half the tips gone it reads Murray’s tree at 3.000, 2.994 and 3.003 across the rules, and keeps the two trees between 0.69 and 0.72 apart.

The repair that worked on a fair loss works on every loss that takes tips one at a time, because a scar does not care why the shoot above it died. The caveat is unchanged too: a scar has to be found, and an overgrown one is a lost tip again.

Why the inner rule matters #

The rules were chosen to bracket what crowns do, and the one that comes out least harmful is the one most often described for trees. Self-pruning of shaded interior shoots is the ordinary way an established crown loses tips, and it is the loss that reads closest to the truth. The rules that read lowest take the outermost tips or the smallest branches’ tips, which is what browsing and drought do to the periphery, and they read about where a fair loss reads.

None of this is a claim that real crowns follow any of the four. It says only that a loss ranked on one property of each tip, applied tip by tip, reads within a few hundredths of the fair loss. The extremes need a loss concentrated in a handful of limbs and sparing their parents, and the measured approximation to that — a limb dying back alone — moves the factor by under two per cent.

What changes for a person counting a crown #

A reading below the built exponent is still the expected damage, and the earlier rule of thumb holds for every rule drawn: a tree that has lost half its tips reads about nine tenths of its exponent, between 0.87 and 0.94 of it depending on which tips went first.

A reading above the built exponent is possible, and should not be read as a third rule. On a crown with patchy dieback — several limbs stripped to a shoot or two while their parents carry on — a count of the tips still there can read Murray’s tree above three. The search shows that it can; the limb measurement shows that one dying limb does not do it by much.

One tree’s reading past about half its tips lost does not identify its rule from the count alone. Counting scars does, and so does a second tree known to have lost the same tips.

The range is a bound, not a census #

Two limits carry over from the search. The range found is at least as wide as the true one, never wider, so the statement that a choice can read the exponent at 3.52 is safe and the statement that it cannot read higher is not made. And the range describes what is possible, not what is likely: the four rules occupy a band a few hundredths wide near the fair loss, and the extremes are reached only by a loss with a very particular shape.

The tree is one Yule draw of fifty junctions and one of two hundred, as before, and the twigs that give the regression its spread are the same twigs on both. The rules rank tips on one property each and lose them independently; a rule that clustered its losses by limb, such as the dieback here, was drawn only one limb at a time.

Readings that would undo it #

With no measurement error, any chosen loss whose two readings are not in the ratio three to two. No choice of half the tips on fifty junctions reading the factor above 1.1, or none below 0.8. The highest choice failing to strip the middle-sized branches at least three times as far as the largest. Any of the four rules’ mean factor reaching one at any level, or a count of scars reading anything but the built exponent under any of them. A single dying limb reading above one on half the limbs or more, or by two per cent or more. Each of these is checked every time the measurement runs.

Still open: a few scars to correct the rest #

A count of scars repairs every loss drawn, but it asks for every scar on every branch, which on a large crown is the expensive part of the work. The factor κ, though, is a single number for the whole tree, and it can be computed on any branch whose scars have been counted: the tips it has, the tips it grew. The next measurement counts scars on a sample of branches only — a few, chosen at random or chosen as the reachable ones, which proved the dangerous sample for the junction fit — estimates κ from them, divides the whole crown’s reading by it, and asks how many branches’ scars a person needs to recover the exponent to a stated precision, under a fair loss, under each of the four rules, and under the choices that read highest and lowest.