Given, you have estimates of:

- Basal area per acre (BAPA, ft
^{2}/ac) - 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, ft^{2})

BA = BAPA / TPA

Using the formula BA = p D^{2} / 4, then the diameter of the tree of average basal area (a.k.a., quadratic mean diameter, D_{q}) is:

D _{q}= [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,

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

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