Tree Top Offset / Lean  Edward Frank
Sept 17, 2005

Bob and John,

I am trying to understand the mathematics of the calculations Bob made to get an average offset or 8.3 feet through the mathematical processing of 1800 tree survey data sets. Please bear with me.

Say you had a plane with a circle centered at the x-y intersection with a radius of Q. Then draw 100 radii lines evenly spaced around the circle. The deviation from the y axis could be calculated for each point as the absolute value of sin theta Q where sin theta was measured as the angle from the y axis. If you added all of these deviations from the y axis and then averaged them you would get an average deviation from the y-axis at all of these angles. As I envision it this number should be proportional to the actual radius of the circle. This proportion should be constant regardless of the radius of the circle. I am sure there is a simple and elegant integral to calculate out the average deviation from y in a circle centered on the y-axis, but my math isn't up to the task. [Could you enlighten me?] What this would mean is that by knowing the average of these deviations, the actual radius of the circle could be calculated.

In Bob's case of 1800 trees, the deviation from the y-axis (perpendicular to the line of sight with the z-axis running through the center of the base of the tree) can be easily calculated for every tree to see how far off the tops are in the x direction along the line of sight from the base of the tree. With 1800 samples, and a reasonable assumption that they the tops are randomly distributed in direction offset from the center of the base of the tree, the average deviation from the y-axis should be proportional to the average offset distance of the top in any direction from the true center of the base of the tree.

Essentially it means by knowing the offset in the x-direction you could calculate the actual average offset of the tops without knowing the amount of offset in the y direction. 

Does what I am saying make sense? Maybe this is what you have already done. How about a little help here guys.

Ed Frank 

RE:  Tree Top Offset / Lean John Eicholz
Sept 17, 2005

Hi Ed,

I am glad to try and clarify the result of the calculation you wish to 
achieve. First let me say you have a clear grasp of the problem and it 
is a more interesting problem than it first appears. By solving it we 
can determine the theoretical average lean of the trees in Bob's 
database, even though this figure is not accurately calculated for each 
tree, by making a statistical assumption that the trees' lean is 
symmetric to the point of measurement. Quite an accomplishment of 
abstraction, it would appear. Thank you for the interesting problem.

Actually, the math is quite simple. The observed offset from vertical 
(as seen from the point of measurement) is proportional to the COSINE 
of the angle. Assume a radius of 1. The average offset would be the 
INTEGRAL of abs(cosine theta) around the circle, divided by the length 
of the circle. We can evaluate this as twice the integral of (cosine 
theta) from -pi/2 to pi/2, divided by 2*pi (the length of the path of 
integration). I apologize if this doesn't make sense, but 
mathematically it is accurate.

The integral of the cosine function is the sine function. We thus 
evaluate the integral as 2*[sin(pi/2) - sin(-pi/2)]/(2*pi). , or 0.6366 
and change.

Using an average observed offset of 8.3 feet yields a projected average 
true offset of 8.3/.6366, or about 13 feet.

I believe we can conclude by saying that the trees measured in Bob's 
database appear to have an average offset of 13 feet from bottom to 
measured tip.

John Eichholz

Re:  Tree Top Offset / Lean Edward Frank
Sept 18, 2005

Bob and John,

The question comes to mind about how Bob calculated the offset and what assumptions were made. Bob if you could jump in and clarify the methodology. A first order approximation would be to calculate the horizontal distance to the base then calculate the horizontal distance to the top. 

Parallax could be safely ignored within the slight left or right offset we would be dealing with. 

There are a couple of issues that need to be factored into the calculations - 1) measuring errors with the laser rangefinder base distances, and 2) radius of the tree.

If the top is measured at a click-over point to obtain the greatest accuracy for the vertical aspect of the measurement, then it is likely that the distance measurement to the base is not at a clickover point. (If you move in or out to get a clickover then the calculations are really screwed.) Thus a reading of 25 yards means the distance to the base is somewhere between the reading of 25 and the clickover to 26. I see no reason that any one portion of this interval should be more likely to occur than any other. Therefore the average error over a large number of shots should be whatever distance of the 25.5 yards would turn out to be minus 25 yards. Therefore if the average rangefinder would clickover at exactly 25 yards, then the average distance for a 26 yard reading not at a clickover point would be 25.5 yards - 25 yards = 0.5 yards or a foot and a half. If the average clickover point (to 25 yards) for point was actually at 24.5 yards, then a large number of readings with random errors would actually average out to be a true 25 yards with an average erro of 0. In your estimation, at what distance do laser rangefinders actually change numbers? At approximately the correct number or at a lower value? That would be a distance that would need to be added to the base horizontal distance to correct for instrument error.

2) When you measure the distance to the base of the tree, you are measuring the distance to the front of the trunk not its center. That is being compared to the distance to top - essentially a distinct point. If the tree was 2 feet in diameter then the bottom distance would be short by 1 foot. To correct for this the cbh readings could be used to determine radius. The radius at the base is larger than at breast height so the cbh numbers would need to be corrected. Also the cbh is generally in inches rather than feet. If the base radius was 10% larger than at breast height, then the correction in feet would be = [cbh/(2)(3.14)(0.9)(12)] roughly about 1/68. For trees without any cbh a set number could be used to complete the calculation, A statement if cbh = null, then cbh = 1.2 (whatever number seems appropriate.).

These do not seem like major amounts of correction, but taken together- the laser is 1/2 yard off, the tree is 3 feet in diameter, then the total amount of correction would be 3 feet - a significant proportion of the total averaged offset of 8.3 feet. For the general average it doesn't matter because the values would be added to the values on one side of the y axis and subtracted from those on the other - provided there actually were equal numbers on each side of the axis. 

The reason for adding these corrections to the calculation would be so that a distribution of where the tops were located with respect to the center of the base of the tree could be made. 

The numbers from conifers could be compared to hardwoods for example...

There are a number of assumptions and estimates being made with these corrections. 1) The average calibration correction for the laser rangefinder is whatever. 2) The base of the tree is a certain percentage larger than the tree at breast height. 3) The general assumption of circularity of the tree trunk. 4) The bottom of the tree was measured from the same point as the top.

The errors associated with making these assumptions are much much smaller than if these corrections were not made at all.

Again I am just brainstorming and trying to figure out what the numbers mean.

Ed Frank

Re:  Tree Top Offset / Lean Bob Leverett
Sept 20, 2005

Lee, Ed, John, Will, Jess, Dale:

The attachment speaks for itself --- I hope. Suggestions? Improvements?


Database Developer and Systems Analyst
Information Technologies

Attachment: Offset Diagram

Re:  Tree Top Offset / Lean Edward Frank
Sept 20, 2005


I got the spreadsheet. I will think about it. My biggest concern is still the calculation of AD, as I explained in a note a couple days ago. 

1) If not at a clickover point then the rangefinder should on average be reading 1/2 yard short along line AE;
2) The radius of the trunk should be added to the length of AE. 

Both would increase the length of the line AE and hence AD. The offset GD would be increased for tops in front of the plane of the trunk base and decreased for those beyond the plane of the trunk.


Re:  Tree Top Offset / Lean Bob leverett
Sept 20, 2005


As a user of the rangefinder back to 1996, I learned to routinely scan the trunk from side to side. I look for a clickover point between the middle and the sides of the trunk, since what we are really doing is going for the center of the tree. Additionally, I work the problem of laser rangefinder clickover for both the crown and the base. One must be conscious of elevation changes shifting forward or backward by up to 3 feet searching for the other clickover, but the elevation change is usually half that. Since I'm seldom on a slope of more than 45 degrees, the vertical correction is usually less than half a foot. In future e-mail discussions, we may want to emphasize these points to the rest of the list, although beyond about 12 of us, I doubt that the others have much interest. For the trunk distance, if I find that I can't get a clickover between the center and the side, I move forward or backward until I do get a clickover. If the clickover is at the center of the bole, I can now use the RD 1000 relascope/dendrometer to get the add-on to the pith. This extra step isn't particularly important for small trees, but trees such as the big tuliptrees in the Smokies that are 4 to 7 feet thick must be handled much more carefully. 

I would add to what you say below - just for clarification. The impact on height determinations using the tangent method (without crown point cross-triangulation) would be to overstate height where AG is less than AD and understate it where AG is greater than AD. In the first case, the baseline is overstated, and in the second case, understated. Interestingly, there are cases where AG = AD, even when GD is of significant magnitude. Tangent users get away with sloppy technique where AG and AD act as radii of a circle through G and D. On occasion, I've been with measurers who lucked out when their results matched mine. They proudly proclaimed their methods of equal efficacy and wondered what all my fuss was about in championing hypotenuse-driven measurements to the actual crown point. None had a clue as to the assumptions each was implicitly making and it proved pointless for me to try to educate them. 


Re:  Tree Top Offset / Lean Edward Frank
Sept 20, 2005


If you measure the position of the top, and then move and measure the position of the base these two numbers are no longer directly comparable. The horizontal offset will change as you move, even if the height measurements are not affected significantly. If you move toward the trunk by 2 feet to get to a clickover point then the relative position of the measured top and the measured base will also be offset by the same amount - 2 feet. The offset can only be measured if the top and bottom positions are measured from the same point. Movement adds or subtracts the amount of lateral movement toward or away from the trunk to the calculated offset position. 

If you move to a base clickover point, that would offset the non-clickover point correction I suggested, but add another correction that would depend on whether you tended to move forward or backward to obtain clickover more often. From a measurement perspective, and I understand the point of the measurements was to obtain height not top offset, the procedure would be better to not move to a click-over point, but accept the average error that convention would entail. This error is a statistically random error that can be averaged over the dataset- the other is a subjective error that is not random and might be applied differently between different people in similar situations.

I need to think about this some more. Your second paragraph is right on the money. It is exactly what it means. To elaborate half of the errors would be greater than the average offset errors. (and half the errors would be less than the average offset errors) I think that the tops measured are more likely to be on the facing side of the tree than on the opposite side of the tree, but for calculation purposes we used to calculate actual offset, this front to back asymmetry is not important. [It could be calculated from the dataset]. Therefore the likelihood of a tree being measured too high, even if the actual top is found would be greater than the chance of it being underestimated. If the measurer is not hitting the top, then almost definitely it would be high. The laser lets you identify the top more clearly than other methods because it generates a number, so the chance of not picking a top, or a correct top is much much greater when just guessing without using a laser. Most of the bigger errors are the result of mis-identifying the top, lesser errors are a direct result of the top not being directly over the base, even if it is identified correctly.


Re:  Tree Top Offset / Lean Bob Leverett
Sept 20, 2005


Yes, of course. You are correct. I changed horses in the middle of the stream as I began reviewing my method for getting ever more accurate heights. Switching to a system where offsets are considered does change the rules. Ed, be damned if I know how we got along before you joined us. I hope others appreciate your contributions as much as I do. I know Will Blozan, Dale Luthringer, and the main measuring crew does.

Your second paragraph is so very important. Oddly foresters more consistently resist this logic than others. The mistakes of top identification are indeed on the measurer's side of the tree far more frequently than on the opposite side where high tops are often not even visible. It is the steep angle of a jutting limb high within the crown that creates the illusion of a top and leads to measurement of false tops. Those who mechanically set base lines of 100 feet so they won't have to do any math with their clinometer are the ones who mis-measure by 50 feet and more. 

I'm always surprised that more professional foresters don't buy themselves cheap laser rangefinders and make good uses of them. Don Bertolette once told me that as a group foresters tend to be weak in conceptualizing mathematical ideas. They do fine running numbers in an add/subtract mode, but are seldom strong as abstract thinkers. Since Don holds a masters in forestry, I gave considerable weight to what he said. He said made that statement years ago and in the intervening time, I've confirmed what he said over and over. 


Re:  Tree Top Offset / Lean Edward Frank
Sept 20, 2005


I guess the tradeoff is more accurate heights, versus better offsets. Which is more important - I would vote for heights.


Shrinking white oaks and cottonwoods    Robert Leverett
   Jan 23, 2006 08:07 PST 


   It is interesting how many white and red oaks have been measured by
others to reported heights of 150+ feet that on measuring them
accurately turn out to be between 100 and 115 and 120 at most. The
architecture of broad-crowned oaks invite a 20 to 40-foot height error.
However, I think that some of the worst over-measurements probably have
come from the open-crown spread of the eastern cottonwood. As we've
discussed on the phone, the cottonwood is a species that has been
reported to 200 feet, yet from our measurements, struggles to reach 135.
The characteristic "V" shape of cottonwood crown invites significant
over-measurement. I keep hoping for a 130 in New England, but haven't
made it yet. I suspect that southern Connecticut has a few in the
130-height class.

   Beyond the broad-crowned oaks and cottonwoods, any thoughts on other
species that "invite" over-measurement?


Will Blozan wrote:

Just for the record, a white oak measured in 1974 to 155' by Arthur
Stupka (of GRSM fame) is now, 32 years later only 115' tall.
RE: Tree top offsets    Robert Leverett
   Feb 01, 2006 09:05 PST 


As I recall, the hardwoods were about 1.5 times the conifers. I'll run
more analysis soon and report the results. With any luck, I'll be back
to measuring trees this weekend and have an experiment in mind.

One point to remember is that the vast majority of the hardwoods in my
database are the forest grown shape because of my preoccupation with
height. I am thinking about shifting gears and doing a much more
rigorous study of crown point offset by species. Goodness knows that I
don't have enough numbers to play with these days.


Edward Frank wrote:

In your calculations of how far the tops of the trees were offset from being
directly over the base of the tree, did the analysis include both
pine/conifers and hardwoods? I would guess that trees with a broader
crown, like maples and oaks, might have a greater average offset from the base than do white pines and spruces. What is your take on the matter? Did you run the calculations to see if this was the case?

Tree Offsets   Edward Frank
  Nov 15, 2006 21:32 PST 


In September 2005, you, John Eicholz, and myself had a discussion about
the offset of the tops of trees. Analyzing 1800 trees you found an
average offset, either toward or away from you of 8.3 feet. If we
assumed that the tree tops were randomly oriented then the average
offset could be calculated. The assumption is that the tops facing you
would be more likely to be measured, but that the tops on the opposite
side would essentially mirror the pattern on the facing side.    

John Eicholz wrote: "First let me say you have a clear grasp of the
problem and it is a more interesting problem than it first appears. By
solving it we can determine the theoretical average lean of the trees in
Bob's database, even though this figure is not accurately calculated for
each tree, by making a statistical assumption that the trees' lean is
symmetric to the point of measurement. Quite an accomplishment of
abstraction, it would appear. Thank you for the interesting problem. The
observed offset from vertical (as seen from the point of measurement) is
proportional to the COSINE of the angle. Assume a radius of 1. The
average offset would be the INTEGRAL of abs(cosine theta) around the
circle, divided by the length of the circle. We can evaluate this as
twice the integral of (cosine theta) from -pi/2 to pi/2, divided by 2*pi
(the length of the path of integration). I apologize if this doesn't
make sense, but mathematically it is accurate. The integral of the
cosine function is the sine function. We thus evaluate the integral as
2*[sin(pi/2) - sin(-pi/2)]/(2*pi). , or 0.6366 and change. Using an
average observed offset of 8.3 feet yields a projected average true
offset of 8.3/.6366, or about 13 feet."

Thinking more about the problem I have some additional input. First I
believe the tree top orientation is really random as was first
suggested. I donít think the measurements are actually random as we
would tend to have better shots at tops pointed generally toward you
rather than off to the side of the vertical position over the base. So
our sample would be biased to favor these shots. I have figured out how
to determine if this is true, but havenít done the calculations yet. I
wanted you and John to comment if possible first. If you look at the
distribution of offsets, you would expect that a plot of offsets from
true vertical would plot as a bell curve with the highest values near
true vertical, in a general flat area near the center, a steeper side,
and a longer tail trailing off at greater offsets.

Think of a plot of the offsets as a series of concentric circles around
the vertical position. The density of the number of the tree tops would
vary from circle to circle as reflected in bell curve shape. Picture
them in 2 foot intervals out to the maximum offset calculated. Now plot
a series of parallel lines perpendicular to the line of sight of the
measurer. Also in two foot intervals these would be tangent to the
circles in a series of points pointing in the direction of the measurer.
OK. The outer circle would only contain points with offsets of some
number of feet, say greater than 26 feet for example. The area beyond
the 26 foot line would only contain points with offsets in the direction
of the observer greater than 26 feet. These two areas would overlap in
an arc segment. All of the data points with values greater than 26 feet
would be found in this arc segment. So you would know that number.
Points on the far side of the vertical would need to be ignored. The arc
segment would represent a certain percentage of the entire area of the
greater than 26 foot half circle. If the distribution of the tree top
offset as measured were really random, then the number of points in each
subsegment of the outer band as divided by the parallel lines would be
proportional to the area of that segment as compared to the area and
number in the known segment. By doing this progressively for each
successively smaller half ring, the number of points contributed by each
true offset could be calculated for every segment in the set. I would
expect, that rather than a bell shaped curve, the central values would
dip down instead. This would reflect that number of longer offsets
would decline as the relative angle to the observer approached 90
degrees. In effect the larger offsets would not be sampled as often in
branches pointed to the sides as they would be in branches pointed
toward the measurer. Thus the sampling was not random.

Now then in a similar manner you could work backward from the center
outward assuming that all of the points in some width of path were from
only those areas within that band. The first segment could be
calculated to be twice the width of the first inner band. Consider the
first segment to only reflect the numbers of offsets of the shortest length
Working outward you could calculate the contributions from all the
different lengths within that band width for each band outward. This
process could be done for bands equal to twice the width of the second
band, and so forth. The overlap of contributions from different lengths
of offsets to each section can be minimized. How to tell which one is
correct from among these choices? First the shape of the sampling
pattern is restricted to those shapes we can calculate. I think a band
of some width pointed toward the measurer is a reasonable shape, but is
not necessarily the only possible shape. But it is one that we can
model. How to tell which width is closest to the truth. We know what
the average of all of the points. I donít have the slightest ideaÖ.

Ed Frank
RE: Tree Offsets   Edward Frank
  Nov 16, 2006 14:09 PST 


Sorry, I meant to send this just to Bob. If you waded through the post,
I am saying if there was a directionality in the sampling of tree top
measurements - the average offset toward the measurer would be the same,
the tree top orientation would still be random, but the integration used
to determine true offset would over-represent the longer distances and
the actual true offset would be somewhat less that 13 feet.
Directionality would just mean that as you circled the tree looking for a
good shot of the top, the best shots might tend to be when the branch
was generally pointed generally toward you as opposed to more to the

The notes I made is a way to test the idea - in science you need to be
able to test your hypothesis and this is how to do it. If it turns out
there isn't directionality involved, or if its effect is minor, then by
adding the numbers in each little box together would give us the actual
distribution of various tree offsets. The latter part was a beginning
of an idea to calculate the pattern of sampling if there was
directionality in the sample... But I lost the train of logic as I was
working through it. I hope directionality is not a significant factor,
as that makes the other calculations simpler.

I guess I am just playing with the numbers to see better understand what
they mean and what hidden assumptions are involved, and what errors are

Ed Frank

RE: Tree Offsets   Edward Frank
  Nov 18, 2006 23:42 PST 

Bob, ENTS,

In the sense I am talking about here directionality does not mean you
are missing tops, or even that there is a flaw in your measuring
technique. It is the idea that branches/tops pointed more toward you
may be slightly easier to see and measure than branches/tops pointed
more to the side. So as you work your way around a tree looking for a
good shot through the clutter, perhaps these angles are choosen more
than others as a consequence.

Ed Frank