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 bigness-fullness
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 semi-perimeter of the
triangle. Then the area A of the triangle is given by the formula:

             A = SQRT[s*(s-a)*(s-b)*(s-c))]

   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 140-foot 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- CBH-35 2, Spread-165, and Height-60

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 three-dimensional 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 non-regular polyon approach that I proposed a few e-mails back
was just to measure projected crown area in a way that allows for
non-regular limb projections - anywhere they occur. We seek to respond
to actual tree geometry. The question is what do we do when we project
the two-dimensional 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 non-regular 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 multi-prism 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 multi-prism shape whether we
were inside foliage or outside, we could adjust the volume of the
multi-prism 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 e-mail.


Bob