As you know, I've passed the derivation of the formula for functional circumference on to Michael Taylor. I have a hard time spotting my own errors. However, here is the derivation with added explanatory comments for anyone else who would take a crack at it.
The formulas in the left block set up the subsequent derivation. For example f and fi, are variables introduced to simplify what follows. My ultimate objective was to express the functional circumference in terms of diameters since the reticle will play a big part in measuring the cross-sectional areas of the stems at the point of separation, with the possible exception of the composite stem. Remember that the underlying strategy, per Zane's approach to volume, is to compute the cross-sectional area of each stem at the point of splitting off the main stem. For two stems, this is a clear process. However, if there are more stems, then the height at the highest split should probably be taken and the cross-sectional areas at that point taken to derive the percentages to apply to the fusion at 4.5 feet (or whatever). Discussion on this point would be helpful.
I see a limited use for this formula. It doesn't replace the case where we judge there to be two or more separate trees. This is meant for a natural coppice such as we see with a fair number of eastern species, cottonwoods, silver maples, box elders, etc. I would include species like pinyon pine, which often exhibit a sprouting form in dry regions.
The formula will greatly discount shrubby forms where sprouting around the root collar accounts for most of the appearance of the shrubby tree. There is an ambiguous case where we may believe that the stems originate from slightly above the root collar so that there is at least a small section of single trunk (maybe only inches). We could then measure the circumference at this point, which would be well below 4.5 feet, but still not involve root collar or limb spread at the narrowest point of the area judged to be a single trunk. Basically, this is what our rules say now.
Don, since you've been doing excellent research into what scientists are discovering about many complex forms such as baobab, me thinks our challenges have only begun for sorting through our options a la big tree contests. Bryant Smith and our other colleagues at America Forests are going to be thrilled. Maybe this can occupy a small part of tomorrow's discussion with the Eligible Species Group.
Robert T. Leverett
Co-founder and Executive Director
Native Native Tree Society
Co-founder and President
Friends of Mohawk Trail State Forest