Piddling
around in dendromorphometry as a cure for insomnia 
Robert
Leverett 
Feb
20, 2007 10:51 PST 
ENTS,
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
piddle.
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
wellknown formula. At other times, though, a truly novel
approach
emerges from considering a measurement problem realistically. As
of
late, I've been preoccupied 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 16foot section of trunk starting at
breast
height from a 100foot 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
crosssectional
area for the log. The average crosssectional area is then
multiplied by
the trunklength 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 midpoint 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 crosssectional 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 100foot trunk a quadratic paraboloid,
then the
diameter, d, at any distance up the trunk, y, would be given by
the
relationship:
d = D*SQRT((Hy)/H), where H = total
height and D = base diameter.
The equivalent
formula for the cone is:
d = (D/H)*(Hy). 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
formula:
V
= H/4*(A1 + A2 + SQRT(A1*A2))
where
V = volume
H
= length of frustum
A1
= crosssectional area of base of frustum
A2
= crosssectional 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 crosssectional 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
Bob
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 
Bob,
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
crosssection
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 
Bob,
In the example below from the middle of your post, you are
talking about the
Huber method in which you are measuring the diameter halfway
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
respectively.
Ed Frank

RE:
Piddling around in dendromorphometry as a cure for insomnia 
Robert
Leverett 
Feb
21, 2007 05:15 PST 
Ed,
I am in total agreement with you about the
utility (or lack thereof)
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 reinvent 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
email, 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.
Bob

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

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

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

Re:
Piddling around in dendromorphometry as a cure for insomnia 
Fores@aol.com 
Feb
22, 2007 14:09 PST 
Bob:
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.
Russ

Re:
Piddling around in dendromorphometry 
Edward
Frank 
Feb
22, 2007 18:44 PST 
Don,
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
offlist, 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 crosssection, 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 
Ed,
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 emails. It could also be part
of the
dendromorphometry book that is in the works.
Bob

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

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,
etc.?
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 foliagethat would really be computationally intense!
Steve Galehouse

Re:
Piddling around in dendromorphometry 
Don
Bertolette 
Feb
23, 2007 20:26 PST 
Ed
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.
Don

Re:
Piddling around in dendromorphometry 
Don
Bertolette 
Feb
23, 2007 20:26 PST 
Steve
Cutting and pasting an example from
http://www.simulistics.com/examples/catalogue/modeldescription.php?Id=branching1
Description
This is a demonstration of Simile's ability to handle fractal
models.
Lsystem (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
buds".
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
r.muetzelfeldt@ed.ac.uk
for more information.
I'm reminded of some inked drawings from the younger Leverett,
of old growth
that aren't that dissimilar from this.
Don

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;
else
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.
Don

Steve
An article that expresses the reasonably long history at
attempting to model
branching, with some interesting images, at...
http://algorithmicbotany.org/papers/abop/abopch2.lowquality.pdf
Don

Fractals 
Edward
Frank 
Feb
24, 2007 19:02 PST 
Don,
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 selfsimilarity. An
object is said to be selfsimilar 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 semicircular rooms had fractal
characteristics. Whether looking at the interlocking pattern of
rooms
in a cave, the semicircular cutouts on the
edges of the room, to the smaller scale semicircular cutouts
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
something.
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 
Ed
Yes, I too was fascinated with fractals for awhile...as I
revisited them, I
was intrigued by the medical applications (neurons, dendritic
pathways,
etc.).
DonB

RE:
Fractals 
Pamela
Briggs 
Feb
26, 2007 08:24 PST 
