New Formula  (Sneak Preview)
  Aug 21, 2007 15:16 PDT 

A new ENTS formula is in the works. At this point, it is simple and may prove very useful to those of us woking with trunk volumes.
Over the past several years of modeling, I've consistently seen that the computing the conical volume of single-stemmed white pines using the cone's base as the area just above the trunk flare (old root collar) and the full height of the tree over-shoots the volume on young and middle-aged pines and understates the volume on old growth trees. I've also observed that the conical volume using the area of the base of the cone as the area at breast height of the trunk along with the full tree height consistently understates the volume. In a sample of 30 modeled pines, the conical volume using breast height area understated the volume in 29 cases. By contrast, the turnk flare base yielded an overstatement in 21 of the 30 cases. For my Taking an average of the two bases yielded a volume that overstated the modeled volume in 12 of the 30 cases and understated 17 and equaled in 1. The average percent of the averaged volumes of the modeled volumes was 98 with standard deviation of 10. Yes,
that's high, but if we introduce a form factor of 1.1 for old growth, columnar forms, and 0.95 for forms that taper extremely fast, and 1.00 for trees that appear just right - a judgment call in each case. Then the average of the ratios of the new volumes versus the modeled volumes remains at 98 and the standard deviation drops to 9. Changing the factors to 1.10, 0.90, and 1.00 yields an average ratio (turned into a percent) of 99 with the standard deviation remaining at 9. SOLD!
The formula needed to estimate the volume of singe-stemmed pines is:

V = H*F*(C1^2 + C2^2)/75.4
V = Trunk volume
H = full tree height
F = form factor (1.1, 1.0, 0.9)
C1 = circumference at trunk flare height
C2 = circumference at breast height.
I'm sure Will, Jess, myself, and others interested in trunk modeling can refine this formula, but even as it stands, it is pretty good. Its algebraic derivation follows for those who wantit.

D1 = diameter at trunk flare height
D2 = diameter at breast height.
C1 = circumference at trunk flare height
C2 = circumference at breast height.
A1 = cross-sectional area at trunk flare
A2 = cross-sectional area at breast height
V1 = trunk volume using cone base area as the area at trunk flare height
V2 = trunk volume using cone base area as the area at breast height.
H = full tree height
F = form factor (1.1, 1.0, 0.9)
V = averaged trunk volume
V1 = H/3*A1
V2 = H/3*A2
V = (V1 + V2)/2
V = (H/3*A1 + H/3*A2)/2
V = H/6*(A1 + A2)
A1 = PI*D1^2/4 = (PI*(C1/PI)^2)/4 = C1^2/(4*PI)
A2 = C2^2/(4*PI)
V = H/6*[C1^2/(4*PI) + C2^2/(4*PI)]
V = H/(24*PI)*(C1^2 + C2^2)
V = H*(C1^2 + C2^2)/75.4 where 24*PI = 75.4
Introducing the form factor F, we arrive at the final formula:

V = H*F*(C1^2 + C2^2)/75.4.

This formula does not replace a full modeling, but should give a pretty good approximation of trunk volume for single-stemmed white pines and I presume other conifers. I particularly like this formula because it does not require the use of either the RD1000 or Macroscope 25. The formula won't apply to most hardwoods. However, hardwoods such as young tuliptrees with a still symmetrical crown may fit well enough.

My current objective is to develop an easy way of obtaining high and low estimates of trunk volume for single-stemmed white pines using simple calculations and avoiding the extra gadgets that some of us use. Once we have the simple method nailed down, we can then search for refinements. As a first cut, the simple formula presented above seems to fit the bill.


RE: Sneak Preview   DON BERTOLETTE
  Aug 22, 2007 04:26 PDT 

Looks like a good first cut may recall that we've had conversations in the past over how badly USFS estimates of old-growth tree volumes are...that's because they have a uniform formula for the broad array of diameters/heights/form factors that they encounter across a broad geographic area.
What you've done, is narrow the population's size diversity, and are now averaging species diversity  within the constrained population of old-growth trees...further refinement could be achieved through similar factors for specific species that typically fall outside the "old-growth un-normal norm"...cypress comes to mind as an obvious 'outlier'
I noticed your use of the word 'judgement (call)'...while it can add confusion to the 'confidence level' when user judgement isn't of 'expert quality', constraining estimates to 'expert quality users' would likely improve quality of estimates.
Back to Don Bertolette
  Aug 22, 2007 10:47 PDT 

   Yes, you're quite right. I saw no other way to get a handle on volume calculations via a simple process than to narrow the field of investigation. I expect most of the eastern conifers, certainly the pines, spruces, and firs, will lend themselves to the outlined modeling process. I'd love to perfect it more in the coming months and then see how well it applies to the ponderosas and other Rocky Mountain conifers next summer. Lots more to do though in the interim.

New Formula
   Aug 23, 2007 19:36 PDT 
Don, Don, Ed, Will, Jess, et. al.

     Today I modeled two more old pines in Childs Memorial Park, Northampton. My sample now consists of 32 pines. Using the formula:

     V = H*F*(C1^2 + C2^2)/75.4

    Where V = trunk volume
               H = full tree height
               F = a choice from the set { 0.90, 1.00, 1.10)
               C1 = girth at trunk/root flare (TF)
               C2 = girth at breast height (BH)

    and modeling the volumes using the Macroscope 25, I got the following results.

Tree                      1             2

Vol-TF             481.6         409.2    
Vol-BH            287.3         307.5
Vol-Avg           384.4         355.3
Vol-Adj Avg    346.0         390.9
Vol-Modeled   350.0         371.3
Adj Avg/MV      0.99           1.05       

    I applied the factor 0.9 to the first tree and 1.1 to the second. The first tree has a large trunk flare that leads to much too high of a volume. The second confirms to the old growth white pine model, so I used 1.1.

    The ratio of the adjusted average to modeled volume (MV) for all 32 sample points is a surprising 0.99 with a standard deviation of 0.09. I can't beat the average ratio value, but the standard deviation could be tighter. However, it is what it is. I could cheat and bring down the standard deviation for the smaple by including a lot of younger trees. But my purpose is to investigate the range of values we can expect using plenty of trees in all ages classes. So far, I'm satisfied with the results and believe the formula has value.

    Tomorrow, I'll model a few younger pines and present the results. Oh yes, the stats on the modeled pines are (120.0, 9.5) and (120.8, 9.7). Despite their closeness in the two common measures, the first has a girth at trunk flare height of 12.3 feet. The second has a girth at trunk flare of 11.3 feet.


New Formula Continued
  Aug 24, 2007 12:10 PDT 

     With respect to the new trunk volume estimating formula, the sample size for testing the trunk volume now consists of 38 trees. To refresh everyone on the formula, which I now refer to as the adjusted averaged trunk volume formula (AATV), the variables and formula are given by:

      V = F*H*/3*(C1^2 + C2^2)/75.4

      where H = full tree height
                C1 = circumference at trunk/root flare point
                C2 = circumference at breast height (4.5 feet above base)
                F = form value (0.90, 1.00, 1.10)

      If we denote the modeled trunk volume by MV, for the 38 sample trees, the current ratio of AATV/MV stands at 0.993 with a standard deviation of 0.088- pretty impressive. The numbers for the unadjusted averaged trunk volume are 0.984 and 0.101 - not bad either. Basically, if there is a large trunk flare and a faster than expected taper, then F = 0.9. If the trunk flare is small to moderate and the taper is very slow, then F = 1.1. All intermediate forms have F = 1.00. This form adjustment requires experience and equally qualified modelers will likely disagree on more than a few trees. So, I hope to replace the form adjustment factor in time, but for the present, it serves a useful purpose. It allows us to compensate for obvious shape exaggerations such as a big trunk flare followed by a quick taper. Not making the adjustment would lead to an over-estimation of trunk volume. On the other hand, a small or modest trunk flare followed by a very columnar form would lead to underest
imating. The use of only 3 factor values is admittedly arbitrary, but so far the three have proven sufficient and avoid the invocation of too many arbitrary form evaluations that the modeler would face. A future refinement will include diagrams showing forms that call for one factor value or another.

      As an example of a spectacular success, the Jake Swamp tree provides a clear case of the process working exceptionally well. Jake's taper is slow, but Jake's trunk flare borders on the high side. With a factor of 1.00, the AATV is 571 versus a MV of 570. That is as good as it gets. However, if I were to introduce a refinement for the Jake tree, I'd probably use an F value of 0.95 to adjust for the "almost too much" trunk flare. This would lead to an AATV value of 542 versus the modeled 570.

      As I see it, the especially attractive features of the AATV formula are that it uses familiar measurements. It employs girth at breast height and full tree height. These are measurements that are normally taken. The measurer need only add one additional measurement - girth at the trunk flare point and then make a judgment call of form by choosing a value for F, which can simply be left at 1.00 if the modeler sees no exaggerated trunk features.

      The AATV formula currently applies only to single-trunked pines. However, I expect it will fit eastern hemlocks, red spruce, loblolly pines, etc. quite well. It won't fit bald cypress. It may also fit single-trunked, narrow-crowned oaks, young tuliptrees, and probably many species of hardwoods grown in close competition with other trees if they maintain narrow crowns. However, additional factors or alternative formulas will ultimately be needed to expand to a wider range of tree forms.

     So, why are we going through this process? Well, the most obvious reason is because we want to. Its what we do, i.e. finding better ways to measure and model trees. Computer methods will ultimately render these simple field techniques obsolete for the experts with plenty of computing experience, but the simper field methods will continue to have a place for both researchers and amateurs, at least for supplying interim measures.

     Well, tomorrow, I will add a couple or three more trees. The time consuming part is modeling with the Macroscope 25. BTW, for the trees being modeled, there is an average 14% reduction in girth going from trunk flare girth to girth at 4.5 feet with a standard deviation of 7%. That is a big variation, but on that is to be expected.

New Formula Cont'
  Aug 25, 2007 20:05 PDT 
Will, Jess, Don, Don, Ed, Lee, et. al.:

   I am up to 41 white pines modeled for testing the new trunk volume fomula. With the inclusion of 3 more pines, the ratio of AATV/MV stands at 0.992 and the standard deviation is 0.089. The unadjusted ATV/MV is 0.982 and the standard deviation is 0.102. Two of the added pines are old growth and one is mature. Adding young pines would improve the ratio and standard deviation, but that is not the purpose. I expect the long run ratio to stabalize around 0.98 with a standard deviation of probably 0.095. We'll see.

   In an experiment, I applied the formula to the big Dunbar Brook hemlock and the AATV fell well short of that modeled. The old growth hemlock form will require an F value of 1.2, if not more. Applying the formula to old growth hemlocks is going to be a challenge.

   Well about an hour ago, I went and did it. I ordered the Macroscope 45. I wanted a backup instrument to the indispensible Macroscope 25 and elected to upgrade to the 45. The telescope version works the same for both models, but the microscope feature has a magnification factor of 45 for the Macroscope 45 instead of 25. One can't have too many gizmos.

   I'm anxious to purchase one of those extremely accurate laser rangefinders that Paul Jost is tracking. The TruPulse 200 is useful, but falls well short of shooting the holes like the Nikon Prostaff 440 does. It is the real work horse of the laser clan.

   While scanning the Ben Meadows catalog, I saw new entries in the one instrument to do everything competiton, but all their tree height routines are the flawed tangent method. So, for now, I'll stick with the simpler instruments and do the math through the ENTS formula package.

RE: Back to Don again
  Aug 26, 2007 05:56 PDT 

   C1 is at the point of trunk/root flare, which is usually from 1.0 to 2.5 feet from ground contact point, at least for eastern trees. C2 is at breast height. I realize that the C1 point is often debatable - a weakness in the method, but not a significant flaw for the vast majority of white pines. It remains to be seen how well the method will hold up for other conifers. For spruce, I think it will be even more accurate, but probably less so for hemlock.


-------------- Original message --------------

You said "I'd say it is safe to conclude that the breast high determination consistently underestimates the trunk volume." By the volume amount due to the flare? Are C1 and C2 circumferences at base and breast height? Seems 'calculable'...

Onward marching volumes
  Aug 26, 2007 10:30 PDT 

    I just finished modeling Monica's tuliptree. I wanted to see how well the AATV formula fit that slender, graceful tree. I believed the match would be close. The rest of you can be the judges. Here are the numbers.

Monica's Tuliptree
CRH       7.95' (Girth at the trunk flare)
CBH       6.60' (Girth at 4.5 ft above base)
Height     123.0' (full height)
V1        206.2 (volume using the cone's base as the area computed at the trunk flare point)
V2         142.1 (volume using the cone's base as the area computed at 4.5 feet above base)    
VC        180.2 (volume modeled with the reticle)
AATV    174.2
Note: AATV = 1.00*123*(7.95^2+6.6^2)/75.4
AATV/VC = 0.97

   This obviously is a very good trunk volume estimation for Monica's tuliptree. Her tree is arrow straight, has a gradual, even taper and a noticeable, but not exaggerated trunk flare. So, I set the F value to 1.00.
    Adding Monica's tree to the sample, I now have 42 modeled trees. My contention continues to be that the formula fits young forest-grown trees very well. Old trees are the rascals. No surprise there, but I think for trees in the 1.0 to 5.0-ft diameter range, the AATV formula will prove valuable. For example, the huge Grandfather white pine in Monroe SF has an AATV value of 976 cubes. Past modeled volumes run from 930 to 1020 cubes, with the most likely value being around 950. Why the variation? Well, it is very difficult to see the trunk in the crown region from the ground. As soon as my toe heals sufficiently and the insect population crashes, it is of the Dunbar Brook I go to re-model the Grandfather pine. I expect to spend the entire day, circling the tree to get more and more points. Hopefully, when Will comes in October, we'll put the issue to rest with a climb.

RE: New Formula Cont'   DON BERTOLETTE
  Aug 28, 2007 13:37 PDT 

Bob/et al-
As I read your post below, realizing how accurate your procedures have become, I couldn't help but think that there probably needs to be different levels of measuring devices for different levels of accuracy needed.
Conceptually, I would suggest there are three levels...that of
2)estimation, and
3)exacting (for the lack of a better word. 
Respectively, these would be purposely for
1) a measurement triage, taken with lightweight field gear [as defined by absence of gear belt/overloaded day- or fanny-pack...;>] such as a clinometer and rag tape;
2) a measure of candidacy for superlative listing, taken with extensive/expensive digital field gear [defined as the array needed to induce field duct tape repairs due to equipment overload and repeated face plants...;>]; and 
3) an exacting, undeniably accurate and precise measure that stands up to the highest scrutiny, and only needed for trees of such superlative dimension that they are likely to be eligible for state or national tree champion status. This third level would necessarily have to be precise (capable of replication by independent measurement) and field going (though not necessarily awkward, but probably so...:>}. 
I'm thinking something that has precise control of vertical and horizontal axis, like a 'Total Station' or such.  There's just too much 'loss of control' in handheld electronics, at the  level of accuracy that ENTS procedures can muster. Stand to the side of anyone measuring the angle to the top and bottom, and you'll see more 'movement' vertically than the accuracy striven for (tenths of a foot).
Re: New Formula Cont'   Edward Frank
  Aug 28, 2007 17:17 PDT 
Don, Bob, et al,

The fourth level of accuracy would be called excruciating....

Ed Frank
Back to Don
  Aug 30, 2007 22:18 PDT 

    I think you are on to something. By recognizing and formalizing the 3 levels of measurement, we would be making a definitive statement about accuracy while making room for those to participate who may be just starting out and intimidated by all the expensive equipment and/or math required to go to the extremes. We would be emphasizing the lesser level of accuracy of Type I measurements and drawing attention to them without rejecting them outright - although Type I measurements would not be ENTS-certified. Let's continue this thread.

    With respect to equipment, LaserTech is delivering a TruePulse 360 to my doorstep this morning at 9:00AM. Way cool! Then the testing will begin.

Re: New Formula Cont'
  Aug 30, 2007 22:42 PDT 
Ed, Don, et al:

    Or exhilarating. The 4th stage is where we call in Will to climb the tree. Then depending on how the climb goes, it is excruciating or exhilarating - maybe both. For us landlubbers remaining on the ground, probably just fascinating.

Comparisons and modeling
  Sep 12, 2007 01:39 PDT 

            Much of our ENTS Internet time is spent discussing the big trees that we've measured and what we want to see. From time to time, we discuss historically significant big trees and often wonder if their reported dimensions are anywhere near accurate. There is much food for thought here that calls in not only reports of the past, but also the concept of bigness.

           Ents who are involved in champion tree programs, directly administering or suppoting them, use the American Forests champion tree formula as the arbiter of tree size when working within the context of the champion tree programs. However, we in ENTS have other systems for measuring and judging tree size. Will Blozan's TDI system is an alternative to the champion tree formula. I prefer it, but there is a place for both systems and I don't deprecate the former.

            Trunk modeling is another method for getting at tree size - at least trunk size for the trees of a compliant form, i.e. where trunk form is distinguishable. But modeling trunk volumes is labor intensive and requires another instrument beyond the standard laser rangefinder, clinometer, and scientific calculator. Laser Tech's dendrometer the RD1000 and/or the Macroscopes 25 and 45 are instruments that can be productively employed in trunk modeling. Several of us are now into trunk modeling in a big way.

           Despite our zeal and Will Blozan's extensive modeling of the great southern hemlocks not withstanding, in most ways, ENTS trunk (and limb) modeling is still in its infancy. Tree forms such as that of the humugus live oaks that our friend Larry is confirming are beyond simple modeling techniques. These giant spreaders need a team headed by the likes of Bob Van Pelt and Will Blozan to do them justice. However, for eastern conifers, like the pines, spruces, and hemlocks, we are getting closer to calculating acceptable trunk volumes with far simpler methods. The fromula that I've now presented a number of times holds increasing promise. I repeat it below.

                                                                       V = F*H*(C1^2 + C2^2)/75.4

                V is volume,
                C1 is circumference at 4.5 feet above the tree's base,
                C2 is circumference at the trunk/root flare,
                H is full height, and
                F is a factor that an be applied if the situation calls for it to compensate for exaggerated trunk forms such as:

                          a. Extreme trunk flare,
                          b. A very columnar form,
                          c. A very fast trunk taper,
                          d. Broken trunk,
                          e A combination of the above.

      We also define:

               V1 as the conical volume based on the full height H and a basal area equal to the area at 4.5 feet above ground level,
               V2 as the conical volume based on H and a basl area equal to the area at the trunk flare point.
               Vm as modeled trunk volume.

       We can state that:

                          a. V1 <=V <= V2
                          b. V1 <= Vm <= V2 in a high percentage of trees and almost always in young to mature trees, but not old trees.
                          c. 0.97 <= average(V/VM) <= 1.03 with almost 100% certainty where the sample of trees ranges over young to old trees..

       So, the formula for V is holding up well and we won't be misrepresenting what it does so long as we clearly state that it provides a means of estimating trunk volume and does not replace modeling methods.

      So what can we glean in the way of information that might give us new insights about trunk volumes and as a derivative - tree bigness? One piece of information that I think is not tree trivia is that provides an easy way to investigate common tree size for comparison purposes to place things into perspective. For instance, in central and northern New England, second growth forests are heavily stocked with trees in the CBH range of 4 to 7 feet and height ranges of 70 to 100 feet. Trees in the size range of 16 to 20 inches DBH and 75 to 100 feet tall are everywhere abundant in regrowth forests and those trees are considered economically mature. Trees in the quoted size range have a trunk volume range of about 50 to 175 cubic feet. A big tree person walking through these young, nondescript forests sees little to get excited about. But with many,many trees in the 75 to 150 cubic foot trunk volume range here in New England's forests, it is easy to lose sight of the fact that the forests are still ecologically immature. Except for a relatively few big tree locations and riparian zones, people are now accustomed to a reduced standard. The big trees are in towns.

       On the big tree sites, white pines and hemlocks with trunk volumes in the 400 to 600 cubic foot ranges stand out as large to very large and an 800-foot cube tree is a monster. Yet an 800-foot cuber is only half the volume of the largest eastern hemlock modeled by Will and Jess in the Smokies. And 1600 cubes, as large as that is, is not even half the size of the biggest tuliptrees in the southern Appalachians and only a quarter of the size of the largest live oaks down Larry's way. Scale and perspective are important.

      Consider the General Sherman Tree in California. Based on BVP's work, we know that it has over 50,000 cubic feet of wood. Obviously the scale for tree size is very wide. Taking a respectible 150-cuber in a regrowth New England forest as an example that satisfies modern timber professionals, it would take 333 New England trees to equal the General Sherman tree. Obviously, scales and ranges are far broader for trees than for the characteristics of people - which I suspect effects the way we make and understand comparisons. A 125-lb adult human male is usually considered small. A 375-lb adult human male is huge. The ratio here is 1 to 3 on weight. Yes, there are super-obese people who weigh 500 to 800 lbs and occasionally over 1,000 lbs, but they are not representatives of the human species that inspire any of us. A 100 lb to 400 lb is about the range we can expect and respect. That's 1 to 4, not 1 to 333. So, people comparisons don't provide us with a broad enough scale to fully appreciate tree comparisons.

       So where is this going? Well, I'm unsure except that when it comes to the comparison game, we need to do a lot of thinking about what is ordinary, unusual, exceptional, and very exceptional when it comes to trees and what we do with the information. The ranges for trees are all over the place and that can obscure what is exceptional at the local level. Sometimes a comparatively little is a actually lot for a species. For instance, a 140-foot tall tuliptree is not exceptional throughout much of the range of that species where growing conditions are good, but 170 feet is. That is a 1 to 1.214 ratio. Now, in terms of volume, a 200 -cubic foot tuliptree is very ordinary, but at the other end of the volume range, we have the 4000-cube monsters. That is a a ratio of 1 to 20 for the same species. So, in terms of comparisons, we need all the help we can get to recognize when to get excited with intra-species and inter-species comparisons. Presently, I get excited when a great white goes over 500 cubes. That's a benchmark of significant size for that species for me, north or south.

      Will, Jess, Ed, Lee, etc.your turn with the comparison ratios.

Re: Comparisons and modeling   Dean Hedin
  Sep 12, 2007 20:56 PDT 
I don't like F. It seems to make the rest of the formula almost meaningless if you can just multiply a constant to everything
to make it come out nice.

In my line a work we call that a "Fudge Factor".

I'll be honest, I haven't measured many trees, but I don't need that experience not to like F.

I understand the difficulty of the problem. You would like to get a very good estimate of volume from a minimal set of measurements.

Such a problem may not have a simple solution (or any).

I know how I would find out. I presume you have a set of data that consist of the "simple measures" along with
data of tree volumes measured in some careful manner and are relatively accurate.

I would then write a genetic algorithm that tries different "formula combinations" and then runs these across the data set as a test for fitness of the formula. This process is repeated many times over with the "formula combinations" crossbred until a formula of required accuracy is found (or not).

It's a brute force method. In other words, let the computer find the best formula (or let it tell you it can't find one).

I would think that one could get good estimate of tree volume for an "isolated" deciduous tree by merely taking a high resolution digital picture of the tree against a clear background (like the sky) in the winter, when the leaves have fallen. So long as a scale is indicated in the image an algorithm could then count up the dark pixels in the image and then estimate the volume.
  Sep 17, 2007 17:43 PDT 

Dale, Will, Jess, Howard, et al:

        On Friday I went to MTSF for a brief period, but long enough to model the Mirror tree, a handsome white pine that stands 156.6 feet tall and has an 11.0-foot girth. It models to 533.0 cubes. The AATV formula yields 516.5 cubes for the Mirror Tree. The difference of 3.3% in volume resulting from use of the formula is acceptable as an estimate of the tree's volume. This morning, I modeled a neighbor's pine, a gorgeous tree 8.1 feet in girth and 120.0 feet tall. My neighbor's white pine yields 232.0 cubes and computes on the formula to 239.1. The difference is 3.1 percent. This afternoon, I modeled a pine on Monica's land. It models to 237.1 cubes. The unadjusted formula gives 244.7 and the adjusted formula gives 232.5.

      I have now modeled 46 pines and applied the formula to the same. The volume yielded by the formula averages 98.3% of modeled volume with a standard deviation of 9.6%. Using the adjusted volumes by applying the shape factor to the situations that I've previously described, the formula yields 99.2% of modeled volume with a standard deviation of 8.6%. The very old trees and trees on steep slopes are the ones that predictably produce the high standard deviation. Nonetheless, The formula is proving its worth as a tool for ENTS to use. The beauty of it is that it gives us a good estimate of volume for single-trunk eastern conifers with very little measuring. However, as it under-estimates the volume of old growth forms, the Seneca Pine's volume would be under-estimated by between 10% and 20% unless the shape factor is applied.

     Dale, will you have any time to experiment with the formula in the coming months? Similarly, Will, Jess, and Howard, will you be able to give it a fair test on conifers in your region? I am especially interested in how the formula would work if tweaked it with one extra measurement taken at halfway up the trunk? A measurement at that point would negate the need for the subjectively applied shape factor, but would require the reticle and finding the midway point of the trunk. That would deter its wide spread usage, but allow us to catalog many more trees.

  Sep 18, 2007 01:32 PDT 

I'm thinking that you may end up with a formula adjustment factor (FAF) that may be effective within species, ie, o-g white pines would have xx.x FAF, whereas hemlocks might likely have a somewhat different one, etc., etc.    As well as age, growth habit (steep slopes, creeping soils, etc.), and other peculiarities affecting base.

RE: Modeling
  Sep 18, 2007 07:30 PDT 

   The formula is as follows:

    H = full tree height,
    C1 = Circumference at root flare
    C2 = Circumference at 4.5 feet
    F = shape factor
    V = volume

    V = H*F*(C1^2 + C2^2)/75.4

    If the difference between C1 and C2 is over 2 feet, you can set F to either 0.95 if the taper is slow or 0.9 if the taper is fast. If the taper is extremely slow as with an OG tree, F could be as high as 1.2 for a normal root flare. Big root flare and extremely slow taper would cancel one another, if you get the idea. You be the judge. Young white pines with out a pronounced root flare have F=1.00.


-------------- Original message --------------
From: Dale Luthringer <>;


Id like to try it out on a couple of pines, and maybe compare it a couple of hemlocks as well. Can you please send me the formula again?

Sorry, Im just starting to get my head above water over here after the busy summer season.

Back to Dale
  Sep 18, 2007 11:05 PDT 

     We will need to experiment with the form factor to cut down on the element of judgement. Making it species specific as Don Bertolette suggests will be required. Hemlocks will most likely have a wider range for the form factor.

      It will be interesting to see if we can get this approach to really work and with John Eichholz coming back on board in the not too distant future, we'll have a good team to work on the process. As it stands now, the formula does work for a limited range of white pine shapes, but that's all I can say for sure.

     Now as to Anthony's Mohawk haircut, well even though he doesn't think he would look handsome in a nice new Mohawk, I think he would be mobbed by the ladies. He'd owe us big time. What do you think?

RE: Modeling
  Sep 18, 2007 14:33 PDT 

   Yes, that's the direction all this seems to be pointed. I think the formula will apply well to spruce in Alaska - hint, hint.


#49 and counting
  Sep 20, 2007 12:34 PDT 
Will, Dale, Howard, Jess, Lee, Don, et al:

    I just returned from modeling white pine #49 - a very mature tree growing near upper Broad Brook. Its stats are as follows:

    Girth at Trunk Flare: 10.3 '

    Girth at 4.5 feet:          9.1'

    Girth at 6 feet:             8.8'

     Total Hgt:                115.3'

     The modeled volume with reticle is 289.5 cubes

     The unadjusted formula volume is 288.9 cubes.

     What can I say? I didn't apply an adjustment because the taper is normal and the trunk flare is under 1.5 feet. This tree is a dream match to the fromula, but of course, other trees aren't. The unadjusted percentage of the average of the formula-calculated volumes to the equivalent averaged modeled volume stands at 98.7% with a standard deviation of 9.7%. The comparable stats for the adjusted formula are 99.3% and 8.4% respectively. Although, I'll continue modeling pines, I think the case for that species has been made. It's mainly a question of tewaking the adjustment factor. Anyone have thoughts on the best way to proceed?

#51, counting and adding species
  Sep 20, 2007 12:45 PDT 


After the success with #49, I added two northern red oaks. One I had previously modeled but had excluded from the data. The other is an black-scarlet hybrid in Monica's front yard. The tree's vital stats are:

    Girth at 1.0 ft = 7.6'

    Girth at 4.5 ft = 6.3'

     Height = 84.0 feet

    The modeled volume is 107.2 cubes. The main trunk holds true to near the top branches, so the overall form is right for formula application. The unadjusted formula volume is 108.6 cubes. Wow! Two tight matches in one day of different species. This is starting to get way cool.

  Sep 21, 2007 13:00 PDT 

   I just modeled a slender white oak across the street from Monica's house. Its stats are:

     Height: 95.9 ft
     CFH:   6.59 ft
     CBH:   4.58 ft

    Its modeled volume is 88.5 cubes and the formula volume computes to 82.0 cubes. The 6.5-cube difference seems a lot, given the prior close matches, but not unexpected.

    I am going to turn my attention to hemlocks now and see if the younger ones conform as well as the younger white pines.

Don Bertolette,

    Don, any chance of applying the formula to some of the straight young to mature spruce in your neck of the woods? My current belief is that spruce should be most compliant.

Back to Don
  Sep 22, 2007 13:48 PDT 

   #54 was modeled an hour ago - a fine black oak in a neighbors yard. The modeled volume was 159.8 cubes. The formula gave 158.9. Since the species is an oak there is a lot of branching, as you would expect. So, what the formula means or produces when applied to such a form, is unclear. The close of numbers match is very interesting. More on this later.

  Sep 23, 2007 11:49 PDT 

I would guess that boles with branching may have more hidden mass/volume than those without, the way that many trees will buttress their limbs for strength...
#54 and #55
  Sep 25, 2007 13:23 PDT 

    Yesterday, I modeled Baaby's Hemlock on Petticoat Hill in Williamsburg, MA ((yes, that's two A's in Baaby). The stats for the tree are:

     CRH: 9.8 '

     CBH: 8.1'

     Height: 127.1'

     The modeled volume of Baaby's Hemlock is 277.3 cubes. The unadjusted formula volume is 271.1. The difference represents 2.2% of modeled volume, which is quite acceptable. The F value in the formula was left at 1.00, because the fast taper of the tree appears to be offset by the slightly high trunk flare (1.7 feet more girth than at 4.5 feet).

     The primary reason Ive modeled Baaby's Hemlock is to establish an ENTS record of significant hemlocks in New England before they succumb to the adelgid. Baaby's Hemlock has lost a lot of foliage and may be past the point of no return. It is significant because its 127.1-foot height makes it the tallest measured so far in the eastern Berkshires. Its age appears to be between 130 and 180 years, probably in the 140-150 range.

      Baabys Hemlock becomes #54 modeled and compared to the calculated volume via the trunk volume formula, which continues to perform well when applied to single-trunk conifers up to 200 years in age and not growing on extremely steep slopes or in excessively wet areas, where trunk flare becomes exaggerated.

      Today, Gary Beluzo and I searched for tuliptrees and I modeled another hemlock along Broad Brook. Stats are:

     CRH = 9.35

     CBH = 8.2

      Height = 117.0

      Modeled Volume = 229.1 cubes

      Adjusted Formula Volume = 228.0 cubes

     Unadjusted Formula Volume = 240.0 cubes

      The tree had a normal root flare, but a very fast taper, so F was set to 0.95.

      The reason the formula works well is that the volume of the trunk of eastern conifers usually falls between two conical forms. The upper limit form is the cone formed by using its basal area equal to that of the trees cross-sectional trunk area taken at the trunk flare point. The lower limit is the conical form with the cones basal area equal to the trees cross-sectional area at 4.5 feet up. In both cases, the cone height equals the full height of the tree. Trees that taper extremely fast or extremely slow and trees that have an extreme trunk/root flare can fall outside the above range. A modest compensating factor can be applied that will capture about 3/4ths to 4/5ths of these trees, but some trees will fall out side the range of +/- 10%. We can eventually derive factors to capture most of eastern conifers and perhaps a lot in the West, at least the Rocky Mountain West.

       To repeat the formula:

       C1 = circumference at trunk/root flare
       C2 = circumference at 4.5 feet above base
       H   = full height of tree
       F = trunk form adjustment factor
       V = trunk volume

       V = F*H*(C1^2+C2^2)/75.4

      The adjusted formula volume averages 99.33% of the modeled volume with a standard deviation of 7.94%. This is after 55 modelings.

New formula
  Sep 25, 2007 16:37 PDT 
Will, Dale, Jess, Don, et al:

      I modeled another white pine today - a slender one on Broad Brook that exhibits the trunk form that works well in the formula. The modeled volume came out to be 147.3 cubes and the formula, unadjusted, yielded 144.7 cubes. The difference of 2.6 cubes is pretty minimal. The Broad Brook pine becomes number 48 modeled with reticle to test the formula. I won't be content until I have 100 pines modeled. My guess is that the unadjusted volume via the formula will be about 98% of modeled volume with a standard deviation of around 10%. The adjusted version will be about 99.5% and 8.5%.