proposal
of a new crown measurement 
Robert
Leverett 
May
30, 2007 10:20 PDT 
ENTS,
I've been thinking about the shapes of
the huge live oaks that Larry
Tuccei Jr. has been measuring in the deep South. Larry's
contorted live
oaks have maximum spreads of 130 to 170 feet, girths of 20 to
30+ feet,
and heights of 35 to 70 feet with the end of the trunk and start
of the
limbs typically occurring between 5 and 15 feet feet. Above 10
to 15
feet these trees are all limbs, and consequently, very different
from
what most of us are accustomed to. Much of a live oak's volume
is in its
complex limb structure as descovered by BVP when the Middleton
Oak was
modeled. In fact, the distinguishing feature of these huge trees
is in
their limbs and the space taken up by the extended limbs. The
conventional big tree formula hints at the size of these
southern
behemoths through the girth and average crown spread dimensions,
but
something more is needed if we are to quantitatively capture
live oak
geometry.
I'd like to propose a new measure for
capturing the bignessfullness
of the live oaks. I recommend that we attempt to capture the
crown area.
This could be done by adopting the concave polygon concept of
crown area
projection. The polygon would be composed of adjacent triangles
that
would be used to measure the crown area projected vertically
onto the
ground. The measurement technique needed to implement this
concept
follows.
The measurer would choose the longest limb
extension and stand under
it, marking the spot. A laser shot to the trunk + 0.5 * diameter
would
measure the first side of the first triangle. The measurer would
then
walk to the next conspicuous limb projection if the area in
between were
fairly well filled  or into the gap between major limb
projections, if
there is a conspicuous gape and mark the inner projection of the
gap.
The measurer would shoot back to the initial spot to get the
second leg
of the first triangle and then to the trunk + 0.5*diameter to
get the
3rd leg. The 3rd leg of the first triangle would form the first
leg of
the second triangle, which would be constructed moving either to
the
right or left, whichever would be easier. But once the direction
were
chosen, the tree would be circled in that direction. The
triangle
process would be repeated all the way around the tree's crown.
The 3rd
leg of the last triangle would be the first leg of the first
triangle.
The resulting figure on the ground would be a concave polygon
constructed of triangles. The figure would not be a regular
polygon.
To compute the projected area covered by the
crown, we would compute
the areas of the triangles and add them together. Assume the
legs of any
particular triangle are denoted by a, b, and c and the area of
the
triangle by A. We define s = 0.5*(a+b+c) as the semiperimeter
of the
triangle. Then the area A of the triangle is given by the
formula:
A
= SQRT[s*(sa)*(sb)*(sc))]
By computing the area of each triangle making
up he concave polygon
and adding the areas together, we would get an approximation of
the area
on the ground shaded by the foliage. We would have its
approximate shape
and area. The more triangles, the better the approximation.
Admittedly,
the method is labor intensive, but it would provide us with a
much
better representation of the tree's "bigness". The
concave polygon area
(CPA) would join the maximum spread (MS), girth (G), and theight
(H) to
provide us better quantitative descriptions of these huge
southern live
oaks.
An even better term to define the concave
polygon would be CPA(n)
where acting as a subscript conveys the number of sides of the
polygon.
The more sides, the more accurate the crown area determination.
Of course, this method could be applied to any
large spreading tree
that we measure. The method certainly applies to our huge
sycamores and
white oaks. I think we would all agres that a large part of the
appeal
for most folks of our largest spreaders stems from the huge area
of
crown coverage. The Pinchot sycamore in Connecticut is a prime
example
of a tree that deserves a better measure of its crown dominance.
The average crown spread of the Pinchot is
around 140 feet based on
the usual technique for computing average crown spread. The
circular
area represented by a 140foot diameter is 15,393 square feet.
How much
would this change were we to use the CPA(n) method? Suppose we
found we
were 1000 square feet off. Would this bother us? I think it
should.
An obvious drawback to the the CPA
measurement is the extra labor
involved in getting the measurements and the impediments to
fully
circling the tree. For instance, property boundaries, fences,
hills,
structures, streets, water, etc. may interfere and prevent the
measurer
from circling the tree at the extreme boundary of the foliage.
However,
we have to deal with impediments when we compute maximum or
average
crown spread and the impediments are exactly the same ones as
for CPA.
If we really want to be masochists, crown
volume might be another
measure, but it is too complicated to approximate without an
order or
magnitude more measurements. My vote would be to adopt CPA at
this time.
It just seems to me that the trees that Larry is taking the time
to hunt
down and document for us deserve as much collective ENTS
mathematical
consideration as the tall pines and tuliptrees that some of us
are
inclined to chase.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

RE:
proposal of a new crown measurement 
Edward
Frank 
May
30, 2007 15:46 PDT 
Bob,
An interesting idea, especially the concept of expressing crown
area
rather than volume. I don't have any comments at the moment.
I would like to elaborate on a concept I suggested earlier 
using the
volumes of regular geometric figures to approximate crown
volume. In a
post in February
http://www.nativetreesociety.org/measure/crown_volume.htm
I suggested:
Crown volume. My ideas on this are to measure
average crown spread,
measure thickness of crown (live crown ratio), and then match
the
general shape of the crown to a series of shape diagrams. You
look in
tree books and they show the typical shape of the tree crown. A
grid of
these with shapes down one side from flat (donut shaped) to
pointy 
open grown pines on one axis. The other axis would be from round
footprint to oval to one sided windswept.
There are several solid geometric volumes that can be used as a
basis
for estimates. I will add diagrams to a website version of this
idea,
but for now I will summarize the core shapes:
Cylinder: volume = (pi) r^2 h, where r = radius, and h = height
of
cylinder
Elliptical cylinder: volume = (pi) a b h, where a = 1/2 major
axis, b=
1/2 minor axis, and h = height
In fact any shape with a known basal area can be converted to a
cylinder
volume by multiplying the base area x height. Your crown area
measure
could be converted to a cylindrical volume easily.
There is a website with an online calculator that will do the
math for
you:
Cylinder: http://grapevine.abe.msstate.edu/~fto/tools/vol/cylinder.html
An interesting variation that may have some application is a
Barrel
Shape: http://grapevine.abe.msstate.edu/~fto/tools/vol/barrel.html
And a cylinder with one domed end:
http://grapevine.abe.msstate.edu/~fto/tools/vol/cyl1domefull.html
The next major shape family is the spheres and ellipsoids.
Sphere: volume = 4/3 (pi) r^3 where r = radius
of the sphere.
Sphere: http://grapevine.abe.msstate.edu/~fto/tools/vol/sphere.html
A sphere is a special case of an ellipsoid in which al three
axis are
equal. A common case in tree forms would be that in which the
two or
three of the axis of the ellipsoid are not equal. (Think of an
egg
shape)
Ellipsoid: Volume = 4/3 (pi) a b c where a, b
and c are the radii of
the major axis.
Ellipsoid
http://grapevine.abe.msstate.edu/~fto/tools/vol/ellipsoid.html
The final family is the conical group.
Cone: volume = 1/3 (pi) r^2 h, where r = radius of the cylinder
base
and h = height of the cylinder
Cone http://grapevine.abe.msstate.edu/~fto/tools/vol/cone.html This
page will allow you to plug in a cylinder with a top radius and
a basal
radius. If the form comes to a point, then the upper radius is
entered
as zero.

As time allows I will provide the volume formulas for more
shapes, but
these basic formulas could be used for many crown volume
estimates right
now. They would require measuring the major and minor axis of
the crown
(max and min spread at right angles) This is done to obtain
average
spread anyway. The height would be the live crown measure from
the
first significant branches to the top of the tree. You have
listed the
height of first major branching as one of your key seven
measurements.
It can be used to calculate live crown ratio. For
open area trees,
photographs could be used to better refine the proper shape to
use to
model the tree volume.
Ed Frank

Back
to Ed 
Robert
Leverett 
May
31, 2007 04:30 PDT 
Ed,
I think that there is room for both crown area
and crown volume
determinations to be reasonably pursued. The projected crown
area
(shadow imprint) is something we can do and area projections of
that
sort have been done for trees like the Angel Oak. But
regardless, of
what we choose as a standard, I am of the opinion that our
largest trees
deserve to be measured, and thus documented, in a wide variety
of ways
and crown volume is certainly one of those ways, although a
computationally chellenging one. Fitting the crown of a tree to
a
standard form is a good start.
Bob

Crown
Measures 
Edward
Frank 
May
31, 2007 14:05 PDT 
Bob,
When you do your crown area calculation, if you have a fat tree
you need
to be sure you measure to what would be the center of the tree,
or else
the sides of the triangle may be upwards of 7 feet or more
short. You
are certainly aware of this but it should be mentioned in a
formal
description of the methodology.
As an example of the crown volume calculation look at the
Audubon Park
Oak on
http://www.nativetreesociety.org/fieldtrips/louisiana/audubon/audubon_park_live_oaks.htm
The Audubon Park Oak CBH35’ 2”, Spread165’, and
Height60’
The branches droop downward so the crown actually extends to the
ground,
rather from the branching point upward. It could be
characterized as
half of an ellipsoid: Volume = (1/2)4/3 (pi) a b c where a, b
and c
are the radii of
the major axis. c = height = 60', a = 1/2 max
spread = 1/2(165) =
82.5', b = 1/2 spread at 90 degrees from max (unknown but say it
was 80%
the maximum spread) = 66'. Then the volume of the crown would
be
683, 892 cubic feet.
The crown area would be roughly 17,100 square feet using those
assumptions, and without your polygonal modeling. It would be
interesting to see what the actual b axis length would be, and
what
number you would derive from your polygonal crown area modeling.
The
"footprint" of the crown is something I think would be
worth measuring
on massive trees such as these.
Ed

RE:
Crown Measures 
Robert
Leverett 
Jun
01, 2007 08:15 PDT 
Ed,
Yes, absolutely. That's why I included ray
length + 0.5*diameter.
I think someone by some method calculated the
cross sectional area
covered by the Angel Oak in South Carolina. On occasion those
kinds of
calculations are performed and reported in newspaper articles,
but the
methodology is never described.
Bob

Re:
Crown Measures 
Beth
Koebel 
Jun
01, 2007 12:53 PDT 
Bob,
I was thinking (look out! the smoke is pouring out of
my ears.) about how to measure the crown volume. I
don't have everything in place but one idea that maybe
you could expand on. I think that you could use right
pyramids with the top facing in to the trunk. The
edges of the base would meet each other until you have
covered the whole tree. How one would do this from
the ground I don't know.
The formula for the volume of a right pyramid is
(area of base) X height/ 3
the base is a polygon so you would need the formula
for that as well and it is, according to Pocket Ref,
page 478,
Area = n ar/2 = nrSQR tan(angle) = nRSQR/sin2(angle)
where a = length of a side(all sides being equal)
n = number of sides
r = distance from the center of the polygon to the
edge of a side
R = distance from the center of the polygon to the
corner where two sides meet.
Pocket Ref also simplifies this by the number of sides
the polygon has on page 478.
sides area
3.........0.4330aSQR
4.........1.0000aSQR
5.........1.7205aSQR
6.........2.5981aSQR
7.........3.6339aSQR
8.........4.8284aSQR
9.........6.1818aSQR
10........7.6942aSQR
11........9.3656aSQR
12.......11.1962aSQR
where a = length of a side when all sides are equal.
Beth 
Back
to Beth 
Robert
Leverett 
Jun
04, 2007 07:17 PDT 
Beth,
Thanks for making the contribution. Please
keep your thinking cap on.
We make progress this way.
Perhaps the biggest drawback to applying the
right pyramid shape is
its regularity. The base is a regular polygon (equal sides) and
that
imposes both two and threedimensional limits that our unruly
trees just
won't respect. Ellipsoids may be a better solution, but even
there, we
impose a regular shape that errant limbs and gaps between limbs
and
multiple limb layers don't respect. However, there is no need to
rule
any form out. We should keep the regular pyramid in our
reperatoire. Who
knows, maybe we'll discover tree pyramid power.
The nonregular polyon approach that I
proposed a few emails back
was just to measure projected crown area in a way that allows
for
nonregular limb projections  anywhere they occur. We seek to
respond
to actual tree geometry. The question is what do we do when we
project
the twodimensional base into 3 dimensions. We might think of
enclosing
the crown by a series of prisms each with a triangular base
representing
a slice out of the nonregular basal polygon used to calculate
projected
crown area. We would then only need to multiply the projected
crown area
by the height of the tree to compute the multiprism crown
enclosure.
This is just a tightening of the reins over the cylinder that Ed
has
proposed as a starting point.
We could also use the triangles making
up the polygon to construct a
series of pyramids, but they wouldn't be right pryamids.
Nonetheless, we
could see where the approach leads. Some of the pyramids will
encompass
too much space and in others not enough. By contrast, the prisms
will
always include too much space. I can't think of any tree that
I'ver ever
seen that would be an exception, although Lombardy Poplars come
close.
Back to the encompassing prism model. If
we could determine at
several randomly chosen points within the multiprism shape
whether we
were inside foliage or outside, we could adjust the volume of
the
multiprism shape accordingly by a simple percentage
calculations. The
key to this method is to be able to determine if we are within
the crown
enclosure or not at each point. That is not easy, but I think I
have a
partial solution to the problem, which I'll propose in a future
email.
Bob

