The equation we use to estimate the weight of a tree: what each symbol means, where its two numbers come from, and the two ways it can be wrong.
We cannot put a living tree on a weighing scale. So we measure three easy things, the thickness of the trunk, the height, and the type of wood, and put them into an equation that gives the weight. The equation used in this course is:
AGB = 0.0673 × (ρ × D² × H)0.976
This page has five panels. Panel A reads the equation one piece at a time, so that every symbol is clear. Panel B shows the equation as a picture. Panel C shows where the two numbers 0.0673 and 0.976 come from: nobody calculated them, they were measured. Panels D and E show the two different ways the equation can be wrong.
Read this once now, and come back whenever a word is unclear. The words from the first page (mean, SD, SE, CV, 95 % range) are used again here without explanation.
An equation can be wrong in two different ways. On this page, orange always means Doubt 1, which comes from the equation, and green always means Doubt 2, which comes from the trees.
a and b were found from a limited number of harvested trees, so the whole line may sit a little too high or a little too low. Whatever the error is, it is the same for every tree we calculate.
Potato shop: the weighing scale is set 20 g too high. Every bag is off by the same 20 g.
Even if the line were perfect, two trees of the same size do not weigh the same. One is hollow, another has heavy branches. Each tree lands a little above or below the line, in its own direction.
Potato shop: the scale is perfect, but each potato has its own weight. Some heavier, some lighter.
| Doubt 1 | Doubt 2 | |
|---|---|---|
| Comes from | the equation (a and b) | the trees (data) |
| Name | parameter uncertainty | scatter |
| Type of error | systematic: same direction for every tree | random: different direction for each tree |
| On the chart | the whole line is a little too high or too low | dots above and below the line |
| Smaller if the equation is built from more harvested trees? (n = the number of trees cut down and weighed to make the equation) | Yes. Divide by √n. With 4 times more harvested trees the doubt is halved. | No. It stays the same. |
| Smaller when we add up the weights of many trees in a plot? (m = the number of trees in the plot) | No. It stays the same. | Yes. Divide by √m. For 4 trees added up the doubt is halved; for 100 trees it is one tenth. |
| Size for our equation | about 1 % of the weight, for one tree and for a whole plot | about 40 % for one tree (CV 37 %, factor 1.43; we round to 40 %); about 4 % for the total of 100 trees |
| Shown in | Panel D | Panel E |
AGB = a × (ρ × D² × H)b with a = 0.0673 and b = 0.976
We will read it from the inside out, using our example tree #406: trunk thickness D = 8.0 cm, height H = 6.0 m, wood density ρ = 0.60 g/cm³.
Think of the trunk as a cylinder. A cylinder holds more wood when it is thicker and when it is taller. Its volume is proportional to D² × H: D is squared because the cross-section of the trunk is a circle, and the area of a circle grows with the square of its diameter. To go from volume to weight we multiply by the density of the wood, ρ. So ρ × D² × H is a rough measure of how much wood the tree has. We call it the size value.
For #406: D² = 8 × 8 = 64. Then ρ × D² × H = 0.60 × 64 × 6.0 = 230.4.
The size value is not a weight in kg; the units are mixed on purpose. It is only a number that grows when the tree is thicker, taller, or made of heavier wood. The next two pieces turn it into kg.
"Raise to the power 0.976" is almost the same as "leave as it is", because any number to the power 1 is itself. The power 0.976 makes the number a little smaller: 230.40.976 = 202.2. (On a calculator: 230.4, then the xy key, then 0.976.)
Why is b not exactly 1? Because big trees are a little lighter than their size value suggests, perhaps because big trunks are more often hollow. The table shows what b = 0.976 does. Going down the table, each size value is 10 times the one above it. If b were exactly 1, each weight would also be 10 times the one above it. With b = 0.976 each weight is only 9.5 times the one above it, because 100.976 = 9.5.
| Size value | Compared with the row above | Weight from the equation | Compared with the row above |
|---|---|---|---|
| 10 | – | 0.64 kg | – |
| 100 | 10 times bigger | 6.0 kg | only 9.5 times bigger (6.0 ÷ 0.64) |
| 1,000 | 10 times bigger | 57 kg | only 9.5 times bigger (57 ÷ 6.0) |
| 10,000 | 10 times bigger | 540 kg | only 9.5 times bigger (540 ÷ 57) |
The last step multiplies by a. This turns the number into kg: 0.0673 × 202.2 = 13.6 kg. That is the weight the equation gives for tree #406.
Where do 0.0673 and 0.976 come from? Nobody can calculate them from physics. They were measured, from 4,004 real trees (Chave and others, 2014). Panel C shows how.
Each dot below is one tree: its size value across, its weight up. The green line is the equation. There are 4,004 dots, as in the real study, but they are computer-made. Trees come in all sizes: a sapling with size value 10 weighs less than 1 kg, a forest giant with size value 10,000,000 weighs about 450 tonnes. On ordinary graph paper (left) this is a problem. If the axes stop at 2,000, as here, more than half of the trees do not fit. If the axes went up to the giants, all the small trees would be squashed into one corner.
So we change the grid. On a log scale (right) each grid step means "10 times more": 1, 10, 100, 1,000, instead of 1, 2, 3, 4. Now the sapling and the giant get the same amount of room, and the equation becomes a straight line. This is why everyone who works with this equation draws it on log paper.
b = 0.976 is the slope of the line: how steep it is. With b = 1 the line would go up exactly one grid step for every grid step across (10 times the size, 10 times the weight). With b = 0.976 it is very slightly less steep: the "9.5 times" of the table in Panel A. By eye you cannot tell 0.976 from 1; it shows up only in the numbers.
a = 0.0673 sets how high the line sits. Read the line at size value 10: the weight there is 0.0673 × 100.976 = 0.64 kg. At size value 100 it is 6.0 kg, at 1,000 it is 57 kg. If a were twice as big, 0.1346, every one of these weights would double and the whole line would move up by one small step, without changing its slope. If a were half, 0.0337, the line would move down by the same step.
The next panel shows how this line is found from measured trees.
The only way to know the true weight of a tree is to cut it down, dry it in an oven, and weigh it. Researchers did this for 4,004 trees in 58 tropical forests. Each dot below is one such tree: its size value across, its measured weight up, both on log scales. When all n trees are in, the computer draws the straight line that passes closest to the dots. The slope of that line is b. Its height, the weight where the line crosses size value 1, gives a. That is all an equation is: the best line through measured dots.
How the computer finds the line. It measures, on the log chart, the vertical distance from every dot to a trial line, squares each distance (so that dots above and below do not cancel), and adds them up. Then it moves the line until this total is as small as possible. This is called least squares.
The fitted a and b are never exactly the true 0.0673 and 0.976, because the dots do not sit exactly on the line. With more trees they come closer.
Try it: set n = 20, press Fast-forward, and look at the fitted a and b. Then set n = 4,000 and do it again. With 20 trees the line can tilt a lot. With 4,000 trees it hardly moves.
In Panel C we found a and b from n trees. Now imagine a second team goes to the forest and harvests n different trees. Their dots are different, so their line is a little different, so their a and b are a little different. A third team gets a third line. Press the button to repeat the harvest again and again. The lines pile up into a fan. The width of the fan shows how unsure we are about a and b. This is parameter uncertainty. It gets smaller as n, the number of trees harvested, gets larger.
This panel uses the same n as Panel C (500 trees per harvest). To change n, use the slider in Panel C. This panel then resets.
The orange box is the one that matters for our tree. It answers: by how many percent does the predicted weight of #406 change from one harvest to the next? Try it: set n = 50 in Panel C and press Fast-forward here; then set n = 4,000 and press Fast-forward again. The percentage falls from several percent to about 1 %. The fan gets narrower by √n: 100 times more trees, 10 times narrower. The real equation was built from 4,004 trees, so for the project we take this doubt as 1 % of the predicted weight.
Doubt 1 looks different depending on who you are. The builders of the equation (the researchers who harvested the 4,004 trees) could harvest again and again, and each time the line would land a little higher or lower, with no pattern. To them Doubt 1 behaves like a random error. That is what the fan above shows.
We are users. We take one finished equation and apply it to every standing tree in our plot. We never harvest a tree, so we can never check whether our line is a little too high or too low. The error is frozen into the numbers 0.0673 and 0.976, and it is the same for every tree we calculate. To us Doubt 1 is a systematic error.
We cannot measure it ourselves, but we can borrow the builders' information. Three ways, from easy to hard:
1 · Use what the builders published. A well-made paper reports not only a and b but also how uncertain they are (their standard errors, or the harvest repeated many times on the computer, as this panel does, using the real 4,004 trees). From these one can work out how much the predicted weight at a given size value would move with a new harvest. For this equation the answer is about 1 % of the weight. That is the number we carry into the project.
2 · Compare with a different published equation. Other researchers built other equations from other trees. If two reasonable equations differ by 5 % for our tree, then our systematic doubt is at least of that order. This is a rough but honest check.
3 · Harvest a few local trees. If a few trees of our species and size can be cut and weighed, we compare their true weights with the equation. A small local harvest cannot replace 4,004 trees, but it can show whether the equation is badly off for our forest.
Then treat it as systematic. When we add up a plot, we apply the 1 % to the total, not to each tree separately, because it does not cancel between trees. In practice this means: draw one pair of a and b per run, and use that same pair for every tree in the plot.
Now suppose we knew the line perfectly. A single tree is still not on it. Two trees with the same size value can have quite different weights: one may be hollow inside, the other may have many heavy branches. The equation gives the typical weight. A real tree lands somewhere around it. How far? On the log scale the spread of the dots around the line is σ = 0.357. In plain numbers: a typical tree weighs between 0.7 and 1.4 times what the line says, and the extreme trees (the outer 5 %) weigh less than half, or more than double. This spread is the scatter. Press Play to harvest many trees that have exactly the size of #406 and see where their weights land.
This band does not get narrower when more trees are harvested. Cutting down 40,000 trees instead of 4,000 would fix the position of the line even better. But each new tree would still land between 0.7 and 1.4 times the line. The scatter belongs to the trees themselves, not to how many trees we measured.
What does make the band narrower? Averaging. Move the "average of m trees" slider. Here m is the number of trees you average. If you report the average weight of m trees of this size, the band becomes narrower by √m: 4 trees, half as wide; 100 trees, one tenth. The hollow trees and the heavy-branched trees partly cancel each other out.