Tree Basal Area from DBH

Basal area is the cross-sectional area of a stem at breast height, and it is nothing more than the area of a circle: $BA = \pi D^2/4$. What makes it central to forestry is not the geometry but the measurement. Of everything you might want to know about a standing tree — its height, its volume, its mass, its crown — diameter is the only quantity you can measure quickly, cheaply and accurately with a tape while standing next to it. So forestry counts in basal area, and every stocking guide, thinning rule, competition index and growth model is written in it.

Breast height is not the same height everywhere. Most of the world measures at 1.3 m. The United States and Canada measure at 4.5 ft, which is 1.37 m. Seven centimetres of stem sounds like nothing, and on a straight bole it nearly is; near a butt swell or a buttress, where taper is steep, it is not. Mixing the two conventions in one dataset puts a small systematic bias into every basal area and a larger one into every biomass estimate derived from it, because the allometric exponent amplifies diameter error by a factor of about 2.4.

Three field rules travel with the measurement. On a slope, measure from the uphill side — the downhill side is further from the root collar and reads larger. On a forked stem, a fork below breast height is two trees and a fork above it is one, and where the fork straddles breast height somebody has to make a call and record it. On a buttressed tropical tree, measure above the buttress and write down the height you used, because a diameter taken at a different height is not comparable to anything.

The instrument matters too. A diameter tape measures girth and divides by $\pi$, so it reports the diameter of the circle having that circumference. For any stem that is not perfectly round — which is every stem — that number is larger than the true mean diameter, because a closed curve of a given perimeter encloses the most area when it is a circle. Calipers averaged over two perpendicular passes estimate the mean diameter instead. The two disagree by a percent or two on a round stem and considerably more on a fluted one. Neither is wrong; using both in one inventory without saying so is.

North American foresters carry the shortcut $BA(\mathrm{ft^2}) = 0.005454 \times DBH(\mathrm{in})^2$, and that constant is simply $\pi/(4 \times 144)$ — the circle formula with the square-inches-to-square-feet conversion folded in. The metric equivalent, $BA(\mathrm{m^2}) = 0.00007854 \times DBH(\mathrm{cm})^2$, is less memorable and gets used less.

Run the equation backwards from a stand's mean basal area per tree and you get the quadratic mean diameter — the diameter of the tree of average basal area. It is not the arithmetic mean diameter and is always slightly larger, because squaring weights big trees more heavily, and the gap between the two widens as the stand's diameter distribution spreads. Every yield table and stand density index is written against the quadratic mean, so substituting the arithmetic mean puts you in the wrong row.