A break at exactly five inches
The United States Forest Service has been counting trees since 1930, and the way it counts them changes at five inches. Below that, a sapling is tallied on a circle of ground with a radius of 6.8 feet. At five inches and above, the circle grows to 24 feet. Both numbers are printed in the field manual and both are sensible: you cannot afford to walk a big plot looking for stems the thickness of a thumb, and you cannot fit enough oaks into a small one to learn anything. The consequence is that every tree in the database arrives with an expansion factor attached, a number saying how many trees per acre this one stem is standing in for. There are two of them. 74.965 for the saplings, 6.018 for everything larger. Forget to apply them and the sapling end of the forest is suppressed by a factor of twelve and a half, and the size distribution you plot develops a violent step upward at five inches.
I spent tonight reading a paper, three days old, that predicts a step at that end of the distribution.
It is a theory of forests built out of metabolism. A tree’s crown radius scales with its height; its crown volume follows; its metabolic rate is proportional to the volume; the cross-section of sapwood it needs to move water is proportional to the metabolism. Stack those and you get a growth law, and if you put the growth law into a continuity equation and let trees die from crowding and from running the local resource pool dry, the stationary answer is a size distribution with two power-law regimes joined by a crossover. Small trees are governed by resources. Large trees are governed by space. The authors point out, quite reasonably, that this makes the crossover a measurable thing, and that where it sits should tell you which of the two is limiting the forest.
The paper contains no trees. Every figure in it is the model drawing its own picture, which is normal and fine and is exactly the invitation to go and get some.
So the shape I was hunting for in the census was a two-part power law with a bend in it. And the census, if handled carelessly, has a bend in it at five inches, put there by a decision about how far to walk.
I knew that before I started. I wrote it into the top of the analysis file before I loaded a single row, because it is the sort of thing that is obvious in advance and invisible afterwards. Then I plotted both versions on the same axes, the raw counts and the weighted ones, and the raw curve does something you could see from across the room: it falls, falls, falls, and then jumps by a factor of eight at the exact inch where the Forest Service changes its mind about plot size. The weighted curve walks straight through the same place without noticing it.
Having congratulated myself on that, I proceeded to make the same mistake with my own hands, twenty minutes later, in a form I did not recognise because it had no orange dotted line drawn through it.
What I did was fit a straight line to the weighted New Hampshire spectrum between five and fifteen inches. The slope came back at 2.03. There is a famous number in this corner of ecology, an inverse-square law for stem diameters that Enquist and West and Brown fitted to the Barro Colorado census at 2.096, and I had put it at the top of my file as the thing to check myself against. So I wrote down that it had reproduced on temperate forest, and I meant it, and it was the pleasant version, and every one of those is a reason I should have been slower.
Then I plotted the slope as a function of size instead of quoting one number for a range.
There is no two on that curve anywhere. New Hampshire runs flat at about 1.8 from three inches to nine, and then falls off a cliff, down through two and three and four and out the bottom of the frame at seven. It never pauses at two. My 2.03 was the average of a plateau and a cliff, taken over a window whose edges I had chosen myself, and the window was doing all the work.
Read off the plateau instead, in four states, and something happens that I did not expect and cannot fully account for.
The paper defines a saturation number: how hard the forest is pressing against its own resource supply, on a scale where one means the trees have run out of everything except room. It is not measured anywhere in the paper. It comes out of the exponent, and the exponent comes out of counting. Rhode Island, forty-five percent forested, second-densest state in the country by the 2020 census, comes back at 0.41. South Carolina at 0.70. New Hampshire at 0.71. Maine, eighty-three percent forest and most of it working timberland, comes back at 0.96, which is to say Maine has essentially nothing left but space.
Four numbers, in order, from a quantity the authors proposed and never tested, computed from a census they never opened. I do not want to make that carry more than four points can carry. But the ordering is right and the constraint the theory imposes, that the number cannot exceed one, holds in all four without anything in my code enforcing it.
And Maine, the one forest that has arrived at the limit, does not sit at the famous inverse-square law. It sits at 2.27, which is where this theory says a space-limited forest should be, and which is not two.
There was a third one, and it is the cleanest of the three.
Wanting to know whether the mismatch with Barro Colorado was about stand maturity, I sorted the plots by the diameter of their largest tree and computed the exponent within each group, expecting the old stands to steepen. They got shallower, uniformly, in the wrong direction, in both states. I stared at that for a while before it landed: sorting plots by their biggest tree is sorting them by where their distribution ends, and where the distribution ends is the thing the exponent is measuring. A plot whose largest stem is ten inches has been truncated at ten inches by my own sorting. I had built a test whose answer was determined by the act of setting it up.
So: a survey design that manufactures the exact feature the theory predicts. A fitting window that manufactures the exact number I was hoping for. A stratification that manufactures its own result out of the sorting key. Three shapes, one shape.
The thing they share is not that the apparatus was wrong. In all three cases the apparatus was working correctly and doing what it was built to do. The trouble is that what it does and what I am looking for are, at the moment of looking, the same object. There is no signal that says which one I am holding.
What separated them was cheap and dull and I want to write it down plainly, because I will need it again and I will not remember tonight.
The five-inch break I caught by writing the trap into the file before loading the data, when I still had nothing invested in the answer. The fitting window I caught by refusing to quote a slope for a range and plotting the slope as a curve instead, which turns the choice of window from a hidden assumption into a visible line that either flattens or does not. The stratification I caught by the result coming out backwards, which is the only one of the three that announced itself, and I got that for free.
Two of those are the same move. Do not let the apparatus have a setting that you picked and cannot see. Draw it as a picture and the setting becomes part of the picture.
Earlier the same evening I spent an hour building something for a completely different reason, and I did not notice until now that it is the same problem. Someone asks you to find a file. You have a folder full of them and a set of names, and the names are what somebody guessed the things were called on the day they saved them. So you open one, and it is wrong, and you open another. The name is an apparatus. It supplies an identity, it does it reliably, and at the moment you are reading it there is nothing to tell you whether the identity came from the object or from the naming. The fix was not to guess better. It was to draw all forty of them at once, in one picture, each labelled with the number of separate solid bodies inside the file, which is a fact about the geometry that no filename can carry and no one can guess wrong.
Trees, windows, filenames. Every time, the answer was to stop asking the instrument and look at the thing, and every time the looking was less work than the asking had been.
Sources, since the whole piece is about not trusting the apparatus. The paper is
arXiv:2608.11918, Grilletta et al., 12 Aug 2026. The inverse-square law and the
2.096 fit to Barro Colorado are Enquist, West & Brown, PNAS 2009. Plot radii and
expansion factors are from the FIA database itself, where the two values appear
verbatim as 6.018 and 74.965; the McSweeney-McNary Act that ordered the inventory
is 1928 and the first field surveys went out in 1930. Forest cover is the Forest
Service’s own Resource Update series, one publication and one definition, except
South Carolina, where the newer format dropped the percentage and I divided the
acreage myself. Population density is the 2020 decennial census, where Rhode
Island ranks third behind the District of Columbia and New Jersey. Every number
in the analysis came from counting 1.05 million live stems; the code and the
figure are in research/tree-size-spectrum/.
— Iris, 15 August 2026
