of a new crown measurement
30, 2007 10:20 PDT
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
oaks have maximum spreads of 130 to 170 feet, girths of 20 to
and heights of 35 to 70 feet with the end of the trunk and start
limbs typically occurring between 5 and 15 feet feet. Above 10
feet these trees are all limbs, and consequently, very different
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
modeled. In fact, the distinguishing feature of these huge trees
their limbs and the space taken up by the extended limbs. The
conventional big tree formula hints at the size of these
behemoths through the girth and average crown spread dimensions,
something more is needed if we are to quantitatively capture
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
This could be done by adopting the concave polygon concept of
projection. The polygon would be composed of adjacent triangles
would be used to measure the crown area projected vertically
ground. The measurement technique needed to implement this
The measurer would choose the longest limb
extension and stand under
it, marking the spot. A laser shot to the trunk + 0.5 * diameter
measure the first side of the first triangle. The measurer would
walk to the next conspicuous limb projection if the area in
fairly well filled - or into the gap between major limb
there is a conspicuous gape and mark the inner projection of the
The measurer would shoot back to the initial spot to get the
of the first triangle and then to the trunk + 0.5*diameter to
3rd leg. The 3rd leg of the first triangle would form the first
the second triangle, which would be constructed moving either to
right or left, whichever would be easier. But once the direction
chosen, the tree would be circled in that direction. The
process would be repeated all the way around the tree's crown.
leg of the last triangle would be the first leg of the first
The resulting figure on the ground would be a concave polygon
constructed of triangles. The figure would not be a regular
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
triangle by A. We define s = 0.5*(a+b+c) as the semi-perimeter
triangle. Then the area A of the triangle is given by the
By computing the area of each triangle making
up he concave polygon
and adding the areas together, we would get an approximation of
on the ground shaded by the foliage. We would have its
and area. The more triangles, the better the approximation.
the method is labor intensive, but it would provide us with a
better representation of the tree's "bigness". The
concave polygon area
(CPA) would join the maximum spread (MS), girth (G), and theight
provide us better quantitative descriptions of these huge
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
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
white oaks. I think we would all agres that a large part of the
for most folks of our largest spreaders stems from the huge area
crown coverage. The Pinchot sycamore in Connecticut is a prime
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
area represented by a 140-foot diameter is 15,393 square feet.
would this change were we to use the CPA(n) method? Suppose we
were 1000 square feet off. Would this bother us? I think it
An obvious drawback to the the CPA
measurement is the extra labor
involved in getting the measurements and the impediments to
circling the tree. For instance, property boundaries, fences,
structures, streets, water, etc. may interfere and prevent the
from circling the tree at the extreme boundary of the foliage.
we have to deal with impediments when we compute maximum or
crown spread and the impediments are exactly the same ones as
If we really want to be masochists, crown
volume might be another
measure, but it is too complicated to approximate without an
magnitude more measurements. My vote would be to adopt CPA at
It just seems to me that the trees that Larry is taking the time
down and document for us deserve as much collective ENTS
consideration as the tall pines and tuliptrees that some of us
inclined to chase.
Robert T. Leverett
Cofounder, Eastern Native Tree Society
proposal of a new crown measurement
30, 2007 15:46 PDT
An interesting idea, especially the concept of expressing crown
rather than volume. I don't have any comments at the moment.
I would like to elaborate on a concept I suggested earlier -
volumes of regular geometric figures to approximate crown
volume. In a
post in February
Crown volume. My ideas on this are to measure
average crown spread,
measure thickness of crown (live crown ratio), and then match
general shape of the crown to a series of shape diagrams. You
tree books and they show the typical shape of the tree crown. A
these with shapes down one side from flat (donut shaped) to
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
for estimates. I will add diagrams to a website version of this
but for now I will summarize the core shapes:
Cylinder: volume = (pi) r^2 h, where r = radius, and h = height
Elliptical cylinder: volume = (pi) a b h, where a = 1/2 major
1/2 minor axis, and h = height
In fact any shape with a known basal area can be converted to a
volume by multiplying the base area x height. Your crown area
could be converted to a cylindrical volume easily.
There is a website with an online calculator that will do the
An interesting variation that may have some application is a
And a cylinder with one domed end:
The next major shape family is the spheres and ellipsoids.
Sphere: volume = 4/3 (pi) r^3 where r = radius
of the sphere.
A sphere is a special case of an ellipsoid in which al three
equal. A common case in tree forms would be that in which the
three of the axis of the ellipsoid are not equal. (Think of an
Ellipsoid: Volume = 4/3 (pi) a b c where a, b
and c are the radii of
the major axis.
The final family is the conical group.
Cone: volume = 1/3 (pi) r^2 h, where r = radius of the cylinder
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
radius. If the form comes to a point, then the upper radius is
As time allows I will provide the volume formulas for more
these basic formulas could be used for many crown volume
now. They would require measuring the major and minor axis of
(max and min spread at right angles) This is done to obtain
spread anyway. The height would be the live crown measure from
first significant branches to the top of the tree. You have
height of first major branching as one of your key seven
It can be used to calculate live crown ratio. For
open area trees,
photographs could be used to better refine the proper shape to
model the tree volume.
31, 2007 04:30 PDT
I think that there is room for both crown area
and crown volume
determinations to be reasonably pursued. The projected crown
(shadow imprint) is something we can do and area projections of
sort have been done for trees like the Angel Oak. But
what we choose as a standard, I am of the opinion that our
deserve to be measured, and thus documented, in a wide variety
and crown volume is certainly one of those ways, although a
computationally chellenging one. Fitting the crown of a tree to
standard form is a good start.
31, 2007 14:05 PDT
When you do your crown area calculation, if you have a fat tree
to be sure you measure to what would be the center of the tree,
the sides of the triangle may be upwards of 7 feet or more
are certainly aware of this but it should be mentioned in a
description of the methodology.
As an example of the crown volume calculation look at the
The Audubon Park Oak- CBH-35’ 2”, Spread-165’, and
The branches droop downward so the crown actually extends to the
rather from the branching point upward. It could be
half of an ellipsoid: Volume = (1/2)4/3 (pi) a b c where a, b
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
the maximum spread) = 66'. Then the volume of the crown would
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
number you would derive from your polygonal crown area modeling.
"footprint" of the crown is something I think would be
on massive trees such as these.
01, 2007 08:15 PDT
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
calculations are performed and reported in newspaper articles,
methodology is never described.
01, 2007 12:53 PDT
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,
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.
where a = length of a side when all sides are equal.
04, 2007 07:17 PDT
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
imposes both two and three-dimensional limits that our unruly
won't respect. Ellipsoids may be a better solution, but even
impose a regular shape that errant limbs and gaps between limbs
multiple limb layers don't respect. However, there is no need to
any form out. We should keep the regular pyramid in our
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
non-regular limb projections - anywhere they occur. We seek to
to actual tree geometry. The question is what do we do when we
the two-dimensional base into 3 dimensions. We might think of
the crown by a series of prisms each with a triangular base
a slice out of the non-regular basal polygon used to calculate
crown area. We would then only need to multiply the projected
by the height of the tree to compute the multi-prism crown
This is just a tightening of the reins over the cylinder that Ed
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.
could see where the approach leads. Some of the pyramids will
too much space and in others not enough. By contrast, the prisms
always include too much space. I can't think of any tree that
seen that would be an exception, although Lombardy Poplars come
Back to the encompassing prism model. If
we could determine at
several randomly chosen points within the multi-prism shape
were inside foliage or outside, we could adjust the volume of
multi-prism shape accordingly by a simple percentage
key to this method is to be able to determine if we are within
enclosure or not at each point. That is not easy, but I think I
partial solution to the problem, which I'll propose in a future