Tree Biomass from an Allometric Equation

Allometry is regression, not physics. $B = a D^{\,b}$ has no mechanism behind it. Nobody derived it from anything. It is a straight line fitted through the logarithms of a scatter of trees that somebody felled, sectioned, dried in an oven and weighed — commonly thirty to sixty trees, of one species, in one region, in one decade. Everything difficult about forest carbon accounting descends from that sentence, and it is worth saying before the arithmetic rather than after.

The exponent lands near 2.4 for a reason that is easy to tell afterwards: mass goes roughly as basal area, which is diameter squared, times a height that itself increases with diameter. That story is a rationalisation of a fitted number, not a derivation of it. The remarkable thing is how tightly the fitted exponents cluster — 2.3 to 2.5 across an extraordinary range of species and climates — which is why metabolic scaling theory keeps being invoked to explain them, and why a published $b$ far outside 2.0 to 2.8 usually means the equation predicts something other than whole above-ground dry mass.

Because it is a fit and not a law, it does not transfer. An equation built on coastal Douglas-fir does not describe interior Douglas-fir: form, taper and wood density all move with site. It certainly does not describe another species, and a single "hardwood" equation applied across a mixed stand is averaging over a two-fold range in wood density. Where the species mix is unknown, a pantropical equation that carries wood density as an explicit term beats a diameter-only one by a wide margin, and one that also carries height beats both.

The place these go badly and quietly wrong is extrapolation beyond the largest tree in the fitting set. Most published fits are built from trees under about 50 cm, because large trees are expensive to fell, awkward to weigh and often protected. Applied at 120 cm, such an equation is answering a question the data never asked, and it usually overpredicts — very large trees hollow out, shed crown, and stop gaining mass in proportion to their girth. The error is not a few percent; it can be a factor. Always check the diameter range behind an equation, and when a tree sits outside it, say so in the body of the report and not in a footnote.

Then there are the units. $a$ is not a pure number. Published coefficients are written for DBH in centimetres and mass in kilograms, so $a$ carries kg per cm$^b$ — its dimensions literally change with the fitted exponent, which is why no single SI expression describes it. Numerically it is the mass, in kilograms, that the equation predicts for a tree of DBH exactly 1 cm. Feed a diameter in inches into a metric $a$ and the answer is wrong by $2.54^{\,b}$, about a factor of 11 at $b = 2.4$: large enough to notice, and small enough that someone might not.

One last trap, invisible and systematic. Fitting in log space and exponentiating back gives the median, not the mean. An unbiased stand total needs a correction factor of $\exp(\sigma^2/2)$; omitting it biases every summed inventory low by a few percent, in the same direction, every time.