Quadratic mean diameter & Reineke's stand density index

Take four trees: 6, 8, 12 and 18 inches. The plain arithmetic mean is 11 inches. The quadratic mean - square each diameter, average the squares, take the square root - is 11.9 inches. The quadratic mean is always the larger of the two, and the gap widens the more variable the stand is.

That is not a mathematical curiosity, it is the point. Growth, competition and volume all scale with cross-sectional area, not with diameter, so the tree that represents the stand's average condition is the tree of average basal area - and that tree is bigger than the tree of average diameter. Every stocking guide, every density index and every yield table is written against the quadratic mean, so using the plain average against them quietly understates your stand.

The second form of the formula, working back from basal area and stem count, is the one to use when somebody hands you a cruise summary. It is the same number by a different road, and this calculator computes both with the same constant so the two roads always meet.

In 1933 L. H. Reineke observed that even-aged stands allowed to grow until they start killing their own stems all converge on the same relationship between how many trees are left and how big they are - and that the relationship holds regardless of age or site quality. Plot the survivors against their average size on log-log paper and you get a straight line with a slope near -1.605. Stand density index is that observation turned into a number: the trees per acre your stand would carry if its quadratic mean diameter were 10 inches.

Its usefulness is that it is age-blind and site-blind. A twenty-year-old stand on good ground and a thirty-five-year-old stand on poor ground can be at the identical point in their competitive lives, and SDI says so where a stem count would not. That is why thinning is prescribed against it - and when the thinning is a commercial one, the way the sale itself is run decides what those stems are worth on the day they move.

The index on its own is a raw number. It becomes a decision when it is read against the maximum for your species - which is a published figure from regional research, not something this calculator will guess for you.

Maximum stand density index varies by species and by region, and the published figures for a single species can differ meaningfully between one research station's work and another's. A default typed into this page would be an assumption a landowner plans a thinning around - and thinning is a decision with a decade of consequences and a contractor's invoice attached.

So the field is empty, and it stays empty until you put your species' figure in it. Get it from your state extension service, from the Forest Service research station covering your region, or from the forester who is going to write the prescription. If nobody can tell you the maximum for your species, that is a useful thing to have learned before you cut.

The percentage bands above are the widely-taught density-management zones, attributed to density-management work published in the mid-1980s and used here as approximate zones rather than as switches. A stand does not become a different stand at 34.9 and 35.1 per cent.

As published, SDI is defined on inches and trees per acre against a 10-inch reference tree. Metric formulations exist, usually anchored on a 25-centimetre reference, and they produce a different number for the same stand.

Offering a unit toggle on this one would hand you a figure you could not compare against any stocking guide you own, and the whole value of the index is that it is comparable. So the density index takes inches and trees per acre, and says so, rather than converting behind your back.