1. Basal area per acre (BAPA, ft²/ac)
2. Trees per acre (TPA, #/ac)

Avery and Burkhart (1994) give a formula for calculating the quadratic mean diameter of a stand based on BAPA and TPA (see middle of page 290).

The derivation of this formula is give below.

The average basal area per tree (BA, ft²)

BA = BAPA / TPA

Using the formula BA = p D² / 4, then the diameter of the tree of average basal area (a.k.a., quadratic mean diameter, D_q) is:

Dq	= [4 BA / p ]0.5
	= [4 (BAPA/TPA) / p ]0.5

BUT, we want D_q in inches, therefore we need to include a conversion from feet (for BA) to inches (for D_q).

THEREFORE,

Dq	= [4 (BAPA/TPA) / p ]0.5 * 12
	= [ (BAPA/TPA) / {(p / 4) / 144} ] 0.5
	= [ (BAPA/TPA) / {0.005454} ] 0.5
	= same as pg. 290