In Bob's interpretation to Zane's
"...discussion of competing forms: (1) a redwood that we interpret to be a main stem plus one or more coppice shoots that have fused to the main stem with a separation at below 1/3rd the height of the main stem, and (2) what we interpret as a single stem that then divides into two stems at 1/3rd or more of the height of the taller stem. As I understand your point, in the case of (1), we expect these coppice stems to increasingly fuse to the main stem and the bark to grow around the assemblage so that the fused region is indistinguishable from a single trunk to the naked eye. Many of the redwood giants are of this form. From this starting point, I understand that you wanted to develop a method to compute a "functional volume" for (1) versus (2), principally to establish parity for big tree competition purposes...
I have several comments:a)
I like the mathematic portion of this and later discussions in that it deals with more and more accurate means of assessing comparative volumes in multi-stemmed trees. With Matt's addition of an array of geometric shapes ranging from columnar to conic, and mimicking the potential shapes that they take when "pushed" into proximity, a further refinement in volume determinations can be made. As well, it provides a simple model of the primary shape of the base, making it relatively easy to see how over time there is tendency for such a "fusion" to result in increasingly circular cross-sections.b)
I'd like to now segue into a more plant physiologically based discussion (here I'm relying on my own under-graduate and post-graduate forestry classes, from "sometime" back, and will eagerly appreciate updates from current concepts in plant physiology!). I'd like to refer you to the online Measuring Guidelines http://www.americanforests.org/wp-content/uploads/2014/12/AF-Tree-Measuring-Guidelines_LR.pdf
, Appendix I, pages 75-81; and the Glossary of Terms, Appendix II, pages 82-85.
Basically there I gathered, from a wide literature search, that most all trees start from a single seed source, extend a single (apical) meristem upwards from ground level, and with each successive annual increment of growth, adds rings concentrically about the apical meristem (after each years increment, the apical meristem becomes "pith"). In subsequent years, injury or other opportunity can reconfigure the meristematic energy to that of a lateral meristematic growth. This may come in the form of a 'coppice' due to an impact injury, or in the case of redwoods, from a burl (itself a response to injury?), or a 're-iteration' usually a significant distance up the tree, in response to upper canopy disturbances.c)
In my mind, this should have us consider the relationship of the "branching mechanisms" to the seed source at ground level, to the somewhat arbitrary human reference point, that of breast height, to Zane's somewhat arbitrary 1/3 Total Height level. Functional in traditional forestry for "normal" sized trees, the breast height gets lost in the often millennial growth of the larger species (thinking in terms of the tree's base, of buttressing, large burls, pathogenic growths, and specific to our discussion, fusions). Which is to say branching: pertinent to ground level; breast height; and in Zane's case specifically, branching at 1/3 of the total height.d)
At ground level, at the time of a seed's sprouting, a tree's singularity is a usually a simplistic determination. We've not the longevity as humans to have observed the hundreds and thousands of annual increments, and Zane's point that we cannot know with certainty whether there was a proximity of seedlings that grew together and over time, or morphed like Matt's paper model (kudos Matt, for a scientifically 'elegant' model!) eventually into a circular cross-section, has merit.e)
We can however operate on the assumption (from Wood Science and Technology discipline) that a tree will grow, in the absence of external stimuli, as near a circular cross-section as possible. And when external input, such as gravity, slope, aspect, solar incidence, and specific to this discussion, obstacles are encountered the tree will respond at the site of such input. Consider the Scenarios I-IV
: Scenario I
, two redwood seedlings one inch apart; Scenario II
, two redwood seedlings one foot apart; Scenario III
, two redwood seedlings ten foot apart; and Scenario IV
, two redwood seedlings forty foot apart. After 2000 years has passed, all other things being equal in Scenario I
, it will likely by impossible to tell whether there are one or two trees without destructive sampling; in Scenario II
, it will still be difficult to discern the 'singleness' of the origin; in Scenario III
, two seedlings growing equally to ten foot diameters will just encounter each other, and as they equally expand to twenty foot diameters, they will first create zones of extrusion to each side of the contact, then as they extend to their full ten foot diameter, they will tend towards a circular cross-section, as the bark of each tree fuses from the base and over time with continued contact up the tree, extend the appearance of bark inclusion, much like that of a tree repairing a wound and eventually disguising the wound; and finally in the case of Scenario IV
, after 2000 years, the two twenty foot diameter trees just encounter each other and it's clear to all that we are looking at two magnificent gob-smacking giant redwood trees.f)
In the Scenarios that involve contact, they respond to that contact with displacement of the circularity expected of trees [in the absence of any other external input], the only difference is the length of time the trees have to "disguise" themselves as one, and the height at which they "emerge from their disguise". g)
In the case of redwoods' and their ability to "re-iterate" some significant distance up the tree, these typically take the form of a coppice, initially growing out (phototrophic?) then up, and while it would be expected that they would be necessarily included in a redwood tree's volume, it's not reasonable to consider reiterating stems in an AF formulaic point assessment.
The means by which one can work chronologically backwards, conceptually from Scenarios IV, III, II, and I involves the "circularity" assumption as described in the online Measuring Guidelines, page 75 in Trunk Development section. External inputs cause the tree(s) to adapt, in what are referred to as adaptive or reactive responses. The cross-sections of such trees demonstrate Reaction Wood. "Angiosperms put tension wood on the uphill side of the tree to pull the tree back up straight [response to change in gravity input], whereas gymnosperms produce compression wood on the downhill side of the tree to push the tree back up straight." (Fritts 1976). Working chronologically backwards, or more to the point, from the top of the tree down, one could expect that the trees would go from essentially circular cross-sections, to (at the point of contact or "fusion") increasingly eccentric cross-sections at the base. The degree to which the bases are eccentric, determines the displacement of the pith line from what would have been a central axis in a circular cross-section.
This requires judgement on the part of the observer, and multiple images or views of the complete circumference of the base, but a better estimate of a tree's singularity or their multiplicity is obtained by delineating the perceived pith lines, than merely making the judgement of what happens 1/3 or more of the way up the tree.
Another way of stating this might be better, that is, for mathematical estimates of a tree's volume or of "fused trees" volumes, Zane's formula is perhaps the best yet. But in terms of plant physiology and wood science, the assumptions referred to here in the online Measuring Guidelines (and included Literature Citations) provide a better determination of single tree versus multiple trees issues than any I have yet seen.
I remain open to and encourage other views, theories, citations...I have made a reasonable search of literature available to me (I don't have extensive availability to Inter-library Loans, etc. here in Anchorage) and would look forward to additional discussions and literature on this topic!