Piddling around in dendromorphometry as a cure for insomnia   Robert Leverett
  Feb 20, 2007 10:51 PST 


As the ENTS gathering at Cook Forest SP nears, Gary Beluzo and I have
been polishing up our presentation on dendromorphometry - the art and
science of measuring trees in the field. Gary has been my silent partner
all along in dendromorphometry. He thought up the term. Will and Jess
are also partners, but they are spending their time in the field these
days doing critically important work. That leaves me to stay at home and

In preparation for the lecture at Cook (It may put all of us to sleep,
so bring an extra pillow), I've been searching for new formulas and
processes that can aid the tree measurer in practicing his/her craft.
Most of my forays into the world of trigonometry and geometry fail to
produce anything of real value. But on occasion a formula pops out of an
algebraic derivation that holds promise for use in the field. The
objective is to make our workhorse formulas as computationally simple as
possible. So, sometimes the result is just a more convenient form of a
well-known formula. At other times, though, a truly novel approach
emerges from considering a measurement problem realistically. As of
late, I've been pre-occupied with trunk volume. What follows is the
result of my piddling.

Forest mensuration often treats the shapes of logs as truncated
paraboloids. Slice the trunk vertically in two halves. The edges of a
slice would follow parabolic arcs for part of the length of the trunk -
perhaps the major part of the trunk, starting at about breast height.
The top section of the tree is often treated more as conical in shape
and the base more neiloid. So, taking the entire trunk, you might
observe the base starting out as a neiloid shape (sides are concave),
then changing to a quadratic parabaloid (sides are convex), to conical
(sides are straight), and maybe back to paraboloid near the tip, but
rarely so. Forest mensuration uses formulas for computing the volume of
sections of a trunk. The sections are often 8 or 16 feet in length.

While the ENTS area of interest isn't log volumes, the modeling of tree
trunks, can use most of the same formulas. What might be an example?
Let's say that the first 16-foot section of trunk starting at breast
height from a 100-foot tall tree measures 4 feet for the bottom diameter
and 3.4 feet for the top diameter taken at 20.5 feet (16+4.5). One
foresty calculation called the Smalian method for log volume would
return this calculation:

16*PI*[((3.4 + 4.0)/2)^2]/4, where PI = 3.141593.

       This formula computes an average diameter from the two end
diameters and then uses the result to compute an average cross-sectional
area for the log. The average cross-sectional area is then multiplied by
the trunk-length segment to get the volume of the trunk segment. The
method assumes a paraboloid form that occupies half the cylinderical
form based on the larger, or bottom, diameter. A slightly better way of
applying this process is to take the actual diameter at 8 feet up from
the starting point, i.e. the mid-point of the segment, which in this
case would by at 12.5 feet up instead of averaging the diameters of the
ends of the trunk segment. The averaging process can understate the
volume, if the trunk is truly a paraboloid. The latter method is called
the Huber method. However, the assumption for both these calculating
methods is that the shape of the trunk segment is that of a parabaloid.
Also, the cross-sectional form is considered to be uniformly circular.

What if the form of the trunk segment is conical instead of that of a
paraboloid? Then, the volume of the section, or frustum, would be
calculated from (16*PI /12)*(4^2 + 3.4^2 + 4.0*3.4) - again, assuming a
uniformly circular shape. The calcualted volumes from these two
processes equate to 172.0 and 172.4 cubic feet respectively.

         The actual trunk forms may not be strictly those of a
parabaloid or a cone. Can some simple test be run to check the form
assumptions? Were the 100-foot trunk a quadratic paraboloid, then the
diameter, d, at any distance up the trunk, y, would be given by the

    d = D*SQRT((H-y)/H), where H = total height and D = base diameter.

        The equivalent formula for the cone is:

     d = (D/H)*(H-y). Remember, y, is from the bottom up, instead of the
apex down.

     These are potentially useful formulas where sections being modeled
are fairly long and curvatures are uncertain.

A section of the trunk that fits neither the paraboloid or cone
assumptions is the first few feet of the trunk. The base to DBH height
is often in the shape of a neiloid (concave sides) because of the root
flare. So if we need the volume of a neiloid frustum, we can use the

            V = H/4*(A1 + A2 + SQRT(A1*A2))

           where V = volume
                    H = length of frustum
                    A1 = cross-sectional area of base of frustum
                    A2 = cross-sectional area of top of frustum

Note the formula for the volume of conical frustum is:

            V = H/3*(A1 + A2 + SQRT(A1*A2)). The only difference is the
factor 1/3 for the cone versus 1/4 for the neiloid. The factor for a
paraboloid would be 1/2.

A potentially usefful formula for the frustum of a cone that has an
elliptical cross-sectional form is as follows:

V = Pi*H/12*(D1*D2 + D3*D4 + SQRT(D1*D2*D3*D4))

where   V = colume
H = length of frustum
Pi = 3.141593
D1 = major axis of base of frustum
D2 = minor axis of base of frustum
D3 = major axis of top of frustum
D4 = minor axis of top of frustum

If the overall form is neiloid, then the formula becomes:

V = Pi*H/16*(D1*D2 + D3*D4 + SQRT(D1*D2*D3*D4))

Ooh, I'm nodding off to sleep. Better stop now. ZZzzzzzzz

Robert T. Leverett
Cofounder, Eastern Native Tree Society
Re: Piddling around in dendromorphometry as a cure for insomnia   Edward Frank
  Feb 20, 2007 15:10 PST 


Reading your post a few things bother me. The calculations of the volumes
of different shapes of trunk segments seem fine. The trunk may vary in
shape in a continuum from cylindrical, to concave outward, to conical, to
concave inward with varying degrees of concavity. So the question becomes,
how can you tell what shape a particular trunk segment represents? How to
do that is the what needs to be determined.

If you look at some of the latest numbers Will compiled for tree volumes you
can see a variety of shapes represented. If a cylinder equal to the
diameter and height of the tree has a value of 100%, then trees measured in
his Tsuga Search Project have varied in percentages from 52.3% to 34.8% for
intact, single trunked trees. The Sag Branch Tulip represents 66% of the
ideal volume. So I don't see how any specific formula, even if idealized
to position on the trunk bottom to top, would be able to represent the
actual variation in trunk shape and volumes.

The only way I can see that is practical is to measure relatively short
segments of the trunk to minimize the variations from ideal. I am not sure
that characterizing these smaller segments as parabolic curves versus simple
conical segments would increase the accuracy of the calculations to a
significant degree. I do like the formula for the elliptical cross-section
conical. To what degree will there be a difference between using this
formula for a section that is only moderately elliptical compared to
treating it as a circular cone of the diameter equal to the average of the
two axis of the ellipse?

Ed Frank

Re: Piddling around in dendromorphometry as a cure for insomnia   Edward Frank
  Feb 20, 2007 19:13 PST 


In the example below from the middle of your post, you are talking about the
Huber method in which you are measuring the diameter half-way between the
ends to determine a better average with the assumption the trunk shape is
paraboloid. If you had that measurement, then I think you would be better
off computing the two smaller sections separately and adding them together.

If you are doing ground based or climb based measurements you could
determine the shape of a segment by comparing how well it fit one shape
pattern or the other by progressively examining the measured diameter with
the diameter measurements taken at the next height above and below it

Ed Frank
RE: Piddling around in dendromorphometry as a cure for insomnia   Robert Leverett
  Feb 21, 2007 05:15 PST 


   I am in total agreement with you about the utility (or lack there-of)
of volume formulas applied to a large section of trunk. The
irregularities are simply too many and assumptions about a parabolic
outline are usually a stretch. I present the formulas more for
completeness of the subject than with any expectation of their wide
spread use by ENTS. The subject is periodically worth investigating and
I'd like to be able show others that we consider traditional forestry
models and search for legitimate applications for them so that it
doesn't appear that we're trying to re-invent the wheel.

   The traditional models tend to work where large numbers of logs are
involved. The law of averages takes over. But applying one of the models
to a long section of trunk can lead to errors beyond what we in ENTS are
willing to accept. In the end, that's why Will climbs the trees.

    I should have issued a caveat to the e-mail, explaining the risks of
assuming a regular geometrical shape over a long segment of trunk.
Still, some of these shortcuts can give us approximations, so we should
still have them as part of our repertory - just used with caution.

RE: Piddling around in dendromorphometry as a cure for insomnia   Robert Leverett
  Feb 21, 2007 05:22 PST 


   Yes, if you have the the 3 measurements, you would be better of using
them all. Again, I was attempting to show (without explaining it) how
the process traditionally works. There has been a ton of work done of
trunk shape and taper and basically the conclusion is that across the
broad sweep of species and forms, there is too much irreglarity to
attempt a one size fits all approach. When giving lectures, I will be
careful to state the caveats and assumptions.

Re: Piddling around in dendromorphometry as a cure for insomnia   Edward Frank
  Feb 21, 2007 14:54 PST 


As I noted in the second post regarding Bob's musings, we could calculate
the "shape" of each tree segment by looking at three successive diameters,
then moving upward one measurement at a time. We could show that some
sections are nearly cylindrical, while others narrow sharply. It would be
interesting to see how the form of a measured tree changed in terms of shape
functions from top to bottom as well as showing the shape changes

For those trees for which we have volumes, we could also calculate the
degree of change that best matches the actual measured volume of the tree of
a given base circumference and height. I am not sure what any of this would
get you, but it could be done without hideous calculations. Maybe it would
be a try it and see what you get experiment?

Re: Piddling around in dendromorphometry as a cure for insomnia   Don Bertolette
  Feb 21, 2007 21:51 PST 

In the perfect world, the shapes are continuous and follow some kind of tree
oriented Fibonacci Code that you and Bob seek. From my experience measuring
thousands of mundane, ordinary trees they'll pretty well fit into a normal
curve, but it's the outliers that usually become difficult to estimate
volume...they'll have sweep or crook, oblong bases, vertical ribs, spirals,
swollen butts, j-hooked bases responding to soil creep/erosion, juvenile
wood accumulations, and all manner of distortions.
It's way technical, but I like my friend's efforts to measure virtual volume
displacement...using high resolution LIDAR to quantify the trees volume...I
think that he/they may arrive at a fairly high level of accuracy/precision,
albeit contrived, technological, and disparate to all the reasons we want to
go out into the woods for anyway...
Re: Piddling around in dendromorphometry as a cure for insomnia
  Feb 22, 2007 14:09 PST 

You have actually hit on one of the greatest fears of older
foresters...emergence of the computer simulation based Power Point generation of resource
managers. It seems that the physical aspects of the forestry profession (or
almost any other job that requires strenuous bodily movement)is a major turnoff
to today's youth and there is growing concern of a future shortage of
trained, qualified and productive technical help.

I do not see the efforts of ENTS as a group of entity being out done by some
future generation of foresters.

Re: Piddling around in dendromorphometry   Edward Frank
  Feb 22, 2007 18:44 PST 


I don't really consider myself a numbers guy, although people have suggested
this in previous emails. Bob and others post these math models of various
aspects of tree measurement. I understand the math fairly well, so I look
at what is posted and provide comments on what has been posted. I look and
see how these math equations are being used, what assumptions are being made
in their application, and how well the model presented fits with real world
trees or forest units.   In the last post for example, Bob cited several
formulas for generic calculations of tree volume with the assumption of
certain shapes for the trunks. I know Bob was aware that these formulas for
whole trees would not be applicable to individual trees, but this was not
included in the discussion, so I mentioned it for completeness of the
discussion. If there are good ideas presented, I support them, and if
there are weaknesses I point them out as well. I wholeheartedly supported
the idea of teh Tree Dimension Index, and have posted many times about the
merits of the Rucker Index.

If there is some contribution I can make, or another way of evaluating the
problem I add my ideas to the mix. Considering the measured volume of a
tree as a percentage of the volume of an ideal cylinder is an idea Will is
playing with in his Tsuga Search Project. We had discussed it some
off-list, but not to the general members of the ENTS discussion list. This
concept seemed applicable to discussion we were having, so I threw it into
the mix.

Bob was talking about the changing shapes of various portions of a tree. I
pointed out a method for calculating the shape of a particular segment of a
tree by using a variation of a formula he mention in the text. (I had
mentioned this before in another context.) I don't know of any utility at
this time for doing this calculation, but it could be done.

Finally I added a new idea to the mix. I suggested that if the volume of a
tree was known to be a certain percentage of an idealized cylinder of the
diameter and height of a tree, then a paraboloid shape could be defined that
had the same basal cross-section, height, and volume as the actual tree.
The idea is not to define a single formula that can be applied to all trees,
but to find the formula for an individual tree, and then to plot the
formulas for other trees of the same species. My thinking is that for a
given species of tree these formulas ( I am not sure exactly how to plot it
at this point) might tend to form a cluster of points on a plot. Different
species would likely have a different cluster center. This would be a way
to characterize the general shape of the trunk for different species along
with being able to calculate the amount of spread of the clusters. It seems
reasonable to me that different species may tend to have different shapes
trunks, after all they have different shaped canopies. This idea could be
tested.   Again I am not sure of the utility of this, but it seems neat and
we should be able to do something with the information.

Anyway, I guess I see myself as more of an enabler to the math people,
rather than being a numbers guy myself. I respond to trip reports, comments
about beauty of trees, almost everything. The contrast I guess is that most
people don't respond to the math discussions at all.

Ed Frank
Back to Ed   Robert Leverett
  Feb 23, 2007 05:07 PST 


   Your comments always provide food for thought. Please continue doing
just what you are doing.

   Basically, I expect to run through a couple dozen formulas/processes
to find one of real practical significance to us. So, a lot of this
stuff will drop by the wayside. One project that I have been thinking of
is the developing of a list (yep, another list) of the most useful
formulas to ENTS that utilize the equipment that we use (laser
rangefinder, clinometer, scientific calculator, tape measure, DBH tape,
prism, GPS Receiver, etc.). We could describe the situations where each
formula is of most use. It would be a way of organizing the wealth of
material that flows through these e-mails. It could also be part of the
dendromorphometry book that is in the works.

Re: Piddling around in dendromorphometry as a cure for insomnia   Don Bertolette
  Feb 23, 2007 19:43 PST 

As I tried to visualize your 'shaping', work by Bill Wilson at UMASS came to
mind, as he worked with modelling tree growth...he was 'branching' out of
Chaos theory, Mandelbrot...the idea being that 'fractal iterations' when
constrained by our 'tree branching' rules could be used to model tree
growth. Most of the Mandelbrot sets I've seen (they're fascinating) have
been two dimensional...I suspect that taking it to the 3rd dimension becomes
computationally intense...
RE: Piddling around in dendromorphometry as a cure for insomnia   Steve Galehouse
  Feb 23, 2007 20:13 PST 

Don, ENTS-

Could these "fractal iterations" of branches then be extended to foliage
size and shape: degree of serration, degree of lobing, compound(ed)ness,

Perhaps the ultimate measure of size of a tree should be the amount of
photosynthetic square footage it presents to to the sun, rather than its
"support system"---but then we would have to factor in the efficiency of
the foliage--that would really be computationally intense!

Steve Galehouse
Re: Piddling around in dendromorphometry   Don Bertolette
  Feb 23, 2007 20:26 PST 


Your use of the words 'perfect cylinder' and the proportion of it, reminded
me of what was going on in my head when I was discussing the LIDAR
approach...LIDAR (Laser Imaging Detection and Ranging) can do a pretty good
job of measuring 'virtual volume', as it sorts out the 'backscatter'...folks
are working with software that effectively is like submerging the tree in a
large graduated cylinder to obtain 'volume', by 'displacement'.
Re 'not being a numbers guy', you had me fooled! I'm REALLY not a numbers
guy!! You both go a long ways towards bringing these conceptualizations to a
level where us 'lay people' can grasp at least some level of it.

Re: Piddling around in dendromorphometry   Don Bertolette
  Feb 23, 2007 20:26 PST 
Cutting and pasting an example from

This is a demonstration of Simile's ability to handle fractal models.

L-system (Lindenmayer system) models are based on the idea that a complex
structure can be obtained by repeatedly applying quite simple rules to a
simple initial stucture. Thus, for example, the branching structure of a
mature tree can be arrived at by applying simple rules to a seedling: rules
such as "replace a bud by an extension to the branch, a side shoot, and two

In the Simile implementation, we use a population submodel to hold the set
of branch segments on the tree, and an association submodel to tell which
segment is connectedto which other segment. As we iterate through time, each
segment gives rise to two daughter segemnts, and thus the structure
develops. Each segment is angled with respect to its parent in 3D: thus, the
resulting structure is in 3D, and could be viewed as such in an appropriate
3D rendering environment.

Currently, the model is not available for download. Contact for more information.

I'm reminded of some inked drawings from the younger Leverett, of old growth
that aren't that dis-similar from this.

And lastly for the night, the following narrative that ties the Fibonacci sequence (if you saw or read DaVinci Code, you may recall it playing a 'role' in the movie) to tree branching, in a more mathematical approach...

Fibonacci is the well known sequence of numbers that describes phyllotaxis - the geometric distribution of petals in circular patterns in plants. The sequence runs 1,2,3,5,8,13 and is constructed by taking the sum of the previous 2 numbers and then adding that number to the sequence and so on. We can use recursion to calculate a Fibonacci number for a given n'th degree in the sequence.

function fibonacci( n )
if ( n == 0 || n == 1)
return n;
return fibonacci( n - 1 ) + fibonacci( n - 2 );

We can then grab a section of the sequence using this function to distribute movieclips to produce patterns such as those appreciated on a pinecone. Check out Gabriel Mulzer's Fibonacci Flowers and while your there have a look at the Recursive Patterns.



An article that expresses the reasonably long history at attempting to model
branching, with some interesting images, at...


Fractals   Edward Frank
  Feb 24, 2007 19:02 PST 


Regarding fractals, there are a number of natural systems that have
characteristics similar to the mathematical concept of fractals. A
fractal is an object or quantity that displays self-similarity.   An
object is said to be self-similar if it looks "roughly" the same on any
scale. The object need not exhibit exactly the same structure at all
scales, but the same "type" of structures must appear on all scales.
When I was working in cave systems in the Bahamas and Puerto Rico, one
mathematician suggested that the semi-circular rooms had fractal
characteristics. Whether looking at the interlocking pattern of rooms
in a cave, the semi-circular cut-outs on the
edges of the room, to the smaller scale semi-circular cut-outs in these
smaller areas, to even finer scale ones in these areas the pattern
remained the same. If you looked at a diagram of any of these different
scale features they would be indistinguishable. In natural systems
there are limits to the extent to which the fractal array will continue.
These are limited by the processes that forms them. For example if the
fractals were driven by tidal processes, then they would tend to not be
larger than the range of the tides. The lower end is similarly limited.
It is an area I wanted to investigate further, but funding was cut, and
I do not have sufficient math to do the work myself.

I know there are fractal functions that resemble trees, but I am not
convinced that there is any relationship between the two forms. And
even if trees were found to have branching in a fractal pattern across a
limited range (mathematical fractals are unlimited) I don't know of how
value looking at mathematical generated fractals would be of any use in
understanding trees. Maybe they could grow them for movies scenes or

Looking a real system that has a fractal pattern within a specific
limited range of scales is useful, because by identifying the limits of
the pattern, you may be able to identify the factor and processes that
are responsible for this limiting boundary value.

Ed Frank
Re: Fractals   Don Bertolette
  Feb 24, 2007 19:48 PST 

Yes, I too was fascinated with fractals for I revisited them, I
was intrigued by the medical applications (neurons, dendritic pathways,
RE: Fractals   Pamela Briggs
  Feb 26, 2007 08:24 PST 

Here are two of my favorite fractals sites: