CROWN RATIO USED AS A SURROGATE FOR FORM
i ._
VOLUME EQUATION
FOR NATURAL LONGLEAF PINE STEMS l-/
Robert M. Farrar, Jr. /
Abstract.-- Including an expression of crown ratio as
a continuous inde endent variable significantly reduced the
variation in a D.s H stem-volume equation for total cubic
feet, inside bark, based on data from felled natural
longleaf pines. A volume difference of about 12 percent is
predicted between trees with 30 percent and 70 percent crown
ratio but with the same D and H. Methods, analysis, and
results are discussed along with examples of application of
the equation.
INTRODUCTION
(2) V = b. + b,(f)(D2H)
f = an expression of bole form (shape).
The "combined-variable" or "D*H" volume
function has been widely accepted as a model for
predicting the volume of tree stems. The usual
form is:
(1) V = b. + b,(D*H)
where
In fact, equation (1) is also at times called a
"constant form factor" equation and in it b,
defines the constant form. But, shouldn't we
consider the possibility of a significant effect
of form? If so, what should the form expression
be?
where
V = tree (or stem section) volume
BACKGROUND
D = tree d.b.h., inches
The historic measures of form such as Girard
form class, form quotients, and form point are
all likely candidates but have a common fault:
all are difficult to use because they either
require measurement of an upper-stem diameter,
either inside or outside bark, or determination
of the height to the estimated crown center of
gravity (form point). Also, Girard form class
has no utility for trees with height less than
about 17 feet. Expensive labor and/or dendrometers are usually required to measure upper-stem
diameters and ocular estimation of crown center
of gravity seems tacitly undesirable.
H= tree height (or stem section length),
feet
hop bl = parameters to be estimated.
This equation is usually fitted via weighted
least squares to homogenize the variance in
volume with the weight commonly being l/D*H.
Often, the r2 will be > 0.99, which suggests
little room for improvement. However, a more
general form of equation (1) given by Spurr
(1952) is:
Is there some surrogate for form? An
obvious candidate'is crown ratio or the live
crown length expressed as a percentage of total
height.
Crown ratio percentage is used by silviculturists as an index of the growth or survival
potential Of a tree since (1) it involves a major
dimension of the tree crown (crown length), (2)
it is probably well-correlated with crown volume,
and, thereby, (3) it represents the photosynthesizing foliage. In general, if the crown ratio
I/ Presented at the Southern Silvicultural
Resea‘Fch Conference, Atlanta, GA, Nov. 7-8, 1984.
Principal Mensurationist, Forestry
2/
Sciences
Laboratory, Monticello, AR, Southern
Forest Experiment Station, USDA Forest Service,
in cooperation with the Department of Forest
Resources and Arkansas Agricultural Experiment
Station, University of Arkansas at Monticello.
429
is 50 percent or greater, the tree is considered
Crown length (or
capable of maximum growth.
height to crown base) is relatively easy and
inexpensive to measure along with total height
when using a clinometer, Abney level, transit,
etc. and a measuring tape or rangefinder.
joined the stem.
These trees were obtained by sampling in a
rectangular distribution of d.b.h., total height,
and crown ratio percentage. The number of sample
trees by d.b.h., total height, and crown ratio
classes is shown in Table 1. As illustrated in
this table, it was difficult to find the combination of short trees for a given d.b.h. and a low
crown ratio.
Developing stem-profile functions for
discrete crown-ratio classes has been useful in
predicting tree product-volumes in unthinned
plantation yield studies for slash pine (Bennett
et al. 1978, Dell et al. 1979) and loblolly pine
FeGcia -et al.
- lggznd for planted (Baldwin
and Polmer 1981) and natural (Farrar 1984)
Generally, lower crown ratio
longleaf pines.
indicated larger stem volume for given D and H.
Dell (1979) emphasizes the promise of crown ratio
in improving tree and stand product-volume estimates and suggests future utility in providing
information on biomass, limb-related quality
considerations, and potential for certain insect
or disease conditions. However, Burton (1980)
did not find a significant discriminating effect
of crown ratio in an analysis of several stem
taper models for planted loblolly pine, although
a trend in the coefficients with crown ratio was
evident for some models. Also, Lohrey (1983)
could find no consistent effect of crown ratio in
an analysis of tree volumes in repeatedly thinned
planted longleaf pine plots when it was included
with D2H and other independent variables.
Analysis
The "observed" tree volumes were used in a
conditioned weighted multiple linear regression
analysis (SAS 1979) employing the full model:
(3)
V = bC + b,(100-CR)
where
+ b2(D2H) + b+loO-CR)(D2H)
V = tree total cubic-foot volume, i.b.,
above a 0.2-foot stump
b. = 0.00535 (observed V of a longleaf
tree with D2H = 0; D = 0 inches
and H = 4.5 feet; based on 10
sample trees)
CR = crown ratio, percent
D = d.b.h., inches
The following analysis was performed on a
felled-tree data base to see if some expression
of crown ratio could be used as a continuous
independent variable and surrogate for form in a
D2H volume equation.
H = total height, feet
Weight = l/D*H.
Note that the variable (loo-CR) assumes the role
of the form variable (f) in equation (2). This
particular formulation was used to make the form
variable analogous to Girard form class since it
was reasoned that a low crown ratio would imply
good form and larger volu e for a given D and H.
Note further that if D1 H = 0, equation (3)
implies that V = b0 + bl(lOO-CR).
This could
present the anomaly of a non-fixed b. if CR is <
100 when D*H = 0.
It turns out that for the
conditioned b. chosen (where D = 0 and H = 4.5),
such young longleaf pines will generally have a
crown ratio of essentially 100 percent and the
potential problem is sidestepped.
METHODS
Data
The data base is described by Farrar (1981).
Briefly,
it consists of 214 felled natural
longleaf pines measured at l-inch taper steps for
d.o.b., d.i.b., and lengths between consecutive
taper steps from a 0.2-foot stump to the zero
Addid.o.b. at the tip of the terminal bud.
tionally, diameters, o.b. and i.b., were measured
at the stump, breast height, and live crown base
along with total height and height to the live
crown base. This measurement procedure was used
to (1) minimize data errors, to (2) obtain a good
description of the shape of the stem, and to (3)
facilitate calculation of stem cubic-foot volumes
Cubic-foot volume,
to integer top d.o.b. limits.
was calculated by assuming a conic section
rus t rum) between taper steps and a conical tip
t$"
section.
Section volumes were summed to obtain
"observedll
total cubic-foot tree volume, i.b.,
above the 0.2-foot stump. Crown ratio percentage
was calculated as [(total height - height to live
crown base)/(total height)](lOO). Height to live
crown base was measured from the groundline to
the point on the stem where the ventral side of
the lowest live limb in the main crown body
In the full-model analysis, b, was not significant at the 5 percent probability level so a
reduced model:
(4)
V= b + b#H) + b+-C&H)was
fitted.
In equation (4) all coefficients were significant.
The calculated probability of observing
a greater absolute "t" ratio due to chance was
0.0001 for each one. Plottings of residuals on CR
and D*H did not reveal any particular trends with
either variable. Adoption of model(4) implies
that we accept that the form variable does not
affect the regression intercept but does affect
the slope.
430
Table 1. - Number of sample trees by d.b.h., total height, and
crown ratio classes.
-------------------------------------------------------------------------D.b.h.
Total Height (ft.)
(in.1
4.6-15 16-25 26-35 36-45 46-55 56-65 66-75 76-85 86-95 s-105
--------------c--------------------------------------------------------------
(K$ Cmwn Ratio
0.6
1.6
2.6
3.6
4.6
5.6
6.6
7.6
-1.5
- 2.5
- 3.5
- 4.5
- 5.5
- 6.5
- 7.5
- 8.5
3
1
1
2
1
3
2
1
2
2
2
1
1
1
2
2
;:: 2;
10.6-11.5
11.6-12.5
12.6-13.5
13.6-14.5
14.6-15.5
15.6-16.5
16.6-17.5
17.6-18.5
18.6-19.5
19.6-20.5
20.6-21.5
21.6-22.5
1
1
1
2
1
1
1
1
Total =
4.6-1.5
?6-25 26-35 36-45 46-55 56-65 66-75 76-85 8t5-g!5 %-lofi
---------F ---------------I_----~---------~31 - 50% Crown Ratio
0.6 - 1.5
1.6 - 2.5
;::: : ;:;
4.6 - 5.5
5.6 - 6.5
6.6 - 7.5
7.6 - 8.5
8.6 - 9.5
9.6 -10.5
?0.6-1t.5
lL6-12.5
t2.6-13.5
13.6-14.5
14.6-15.5
15.6-16.5
16.6-1.7.5
17.6-18.5
18.6-$9.5
19.6-20.5
x).6-21.5
27.6-22.5
4
3
1
1
1
2
2
1
1
2
2
1
2
1
1
1
1.
3
1
1
1
P
1
t
1.
1
1
3"
1
z
2
1
1
1
1
2
1
1
1
4
1
2
1
1
1
1
1
1
1
1
Total = 8fj
431
37
Table 1. -
(Cont.) Number of sample trees by d.b.h., total height, and
crown ratio classes.
_______-----------------------------------------------------------------------D.b.h.
(in.1
Total Height (ft.)
4.6-15
16-25 26-35
36-45 46-55
56-65 66-75
76-85 86-95
_______----------------------------------------------------------------------2
- I.5
- 2.5
- 3.5
- 4.5
- 5.5
- 6.5
- 7.5
7.6 - 8.5
8.6 - 9.5
9.6 -10.5
10.6-11.5
f1.6-f.2.5
12.6-13.5
T3.6-14.5
14.6-15.5
15.6-16.5
16.6-17.5
17.6-18.5
18.6-19.5
19.6-20.5
x.6-21.5
21.6-22.5
0.6
1.6
2.6
3.6
4.6
5.6
6.6
51%
Crown
Ratio
3
1
2
1
96-l@
1
1
:
3
4
1
1
1
2
3
2
2
1.
2
2
2
1
1
1,
2
2
1
2
2
3
1
2
1
1
2
1
1
1
1
1
Total = 92
RESULTS
Equation (5) is shown graphically in Figure
for three contrasting crown-ratio perc.entages.
Note that, as expected, a low crown ratio results
in a larger tree volume for given D and H and
implies better stem form for the shorter tree
crown.
Equation (6), if plotted, would nearly
coincide with the line for 50 percent crown ratio
in Figure 1 because the mean CR for the data set
is also about 50 percent.
1
The fitted form of equation (4) is:
(5) V-o.co535 + 0.C018XW2(D2H) + O.oxcc664895 (lcCrCR)(I?H).
For comparison, a conditioned weighted fit of
model (1) was made to obtain the simpler
"combined variable" equation:
(6)
V =O.cOgE + 0.co21Y7~(D2H)
withweight=
l/D2H.
To illustrate the effect of CR on tree
volume, assume we have a tree with D = 10 inches,
H = 70 feet and let CR be 30, 50, or 70 percent.
Calculating the volume for these three conditions
using equations (5) and (6) we obtain:
"Goodness of fit" statistics for equations
At first
glance, these statistics do not indicate an overwhelming superiority by either equation. Both
equations are relatively unbiased as indicated by
nearly zero mean deviations (d) and both account
for a very large proportion of the variation in
tree volume as indicated by the fitaindices (FI)
of about0.99 or larger. Both the relative (%d)
and absolute relative (RMS%d) deviations are
similar for both equations. But, for equation
(5) the root mean squared deviation (RMSd) is
lower by about 19 percent, indicating that the
average absolute size of errors is less with
equation (5).
(5) and (6) are given, in Table 2.
For (6), V = 15.385
For (5) and CR = 30%, V = 16.286
1,
I,
I,
w w 50%, v = 15.355
,I
1,
,I
,a - 70%,
V = 14.425
Compared to equation (6), equation (5) predicts
about 1 cubic foot or 5.9 percent more volume for
CR = 30% and about 1 cubic foot or 6.2 percent
less volume for CR = 70%. The cubic-foot dif-
432
Table 2. --
“Goodness of fit” statistics for equations (5) and (6).
---------------------------------------------------------------------------
Equation
1/
Related
21
--
Statistics
-21
4/
b 2/
6/
Y
d
RMSd
RMS%d
FI
--------------------------------------------------------------------------- - - - - - ft.3,i.b. ---s-s - - - - - - - %- - - - - - - n
(5)
214
17.331
0.000
1.654
-3.404
(6)
214
17.331
0.000
2.042
-2.98
15.139
13.912
7/
0.991
0.987
n = Mumber of observations
P= Mean observed total cubic-foot volume, i.b., per tree
d = Mean deviation = (l/A) f (Yi - ;i)
where Yi = observed
volume of the ith tree and ‘ii = predicted-volume of the ith tree
RMSd = Root mean squared deviation = til/n)5(Yi - +i12
%d = Mean percent deviation = [(l/n)2 ((Y i’ ~i)/yi)l(r'OO)
RMS$d = Root mean squared percent deviation
= til/n)f((yi - ci)/Yi)2 llOo)
FI
= Fit index = 1 - [Z.(Yi - ~i)2/f (Yi - Y)2]
Fit index is analogous to the coefficient of determination
in regular regression and takes on values < 1. FI is used to
evaluate variation accounted for by conditioned or transformed
regressions or systems of equations where the coefficient of
determination is not appropriate.
ferences increase as tree size increases, as
illustrated in Figure 1, but the percentage differences remain essentially the same.
The
difference between the volume of two trees having
the same D and H but a CR of 30 percent for one
and 70 percent for the other is about 11 to 13
percent, depending on the base. This relative
difference is on the order of the 10 percent
volume difference reported by Dell (1979) between
high and low crown-ratio stem-profile functions
for planted loblolly pine.
interpretive.
The trees selected in the rectangular distr,ibution were required only to be
naturally regenerated, single-stemmed, visibly
undamaged, and to fit into one of the D-H-CR
They came from a wide variety of
cells.
undescribed stand conditions and histories
regarding age, site quality, density, and past
treatment.
Therefore, they are not associated
with any particular set of stand conditions or
management regimes. The relationship between
crown ratio and stem volume observed under
specific thinning, regimes over a rotation might
be somewhat different quantitatively from that
herein but should be similar qualitatively if
thinning intervals are long enough for crown
length and stem form to adjust to the new environment.
DISCUSSION
Equation (5) shows that an association
exists between tree volume and crown ratio in
natural longleaf pines, as sampled in this data
base, but any cause-effect relationships are
Basically,
for species exhibiting excurrent
or monopodial growth form, a tree stem tends
433
Cubic feet
per tree
10
CR %
D2H
Figure 1. -- Predicted total cubic-foot volume, i.e., by crown-ratio (CR)
class, equation (5).
toward a cylinder in shape below the crown base
and tapers strongly within the crown. Thus, as
crown lengthens, stem form tends to be poorer.
Stem shape appears to be modified secondarily by
stand density. Open-grown trees generally tend to
taper more in the lower stem than trees in a
closed stand due apparently to a buttressing
response to wind loading. But, also, crown ratio
and stand density are co-related since open-grown
trees also tend to have high crown ratios while
trees in dense stands tend to have low crown
ratios. In the normal development of a stand,
the trend with time seems toward lower crown
ratios and perhaps an approach toward an
asymptote. For the most part, it would seem that
any treatment that altered the crown ratio would
also result, sooner or later, in altered stem
form, but how quickly crown ratio changes and
stem form adjusts to changed environment in
longleaf pine is open to investigation.
crown-ratio effect was inconsistent, the crown
ratio and stem form never achieved commensurate
values or crown ratio never varied consistently
among the study treatments due to frequent thinning. Also, possibly the model form or inclusion
of other co-related variables prevented detection
of an effect of crown ratio. Nevertheless, as
Lohrey points out, lateral crown expansion and
increased loads on the stem due to wind and
increased crown volume after thinning may affect
stem form and volume without an appreciable
change in crown ratio. The point is also well
taken that if we wish best to measure stem form
and volume responses to intensive treatments,
such as frequent thinnings, live prunings, fertilizations, etc., we should resort to equally
intensive
measurements. This implies precise
dendrometry of the stems in stands, or a sufficient sample thereof, plus establishment of
inside- and outside-bark diameter relationships
along the stem.
Perhaps in Lohrey's (1983) report, where a
434
Although crown ratio may not be the most
sensitive expression of stem form, it should be
adequate in many cases. For applications where
the expense of dendrometry is not justified, the
added precision of a D2H volume regression or
stem-profile function employing crown ratio
should prove useful.
Feduccia, D. P., T. R. Dell, W. F. Mann, Jr., T.
E. Campbell, and B. H. Polmer.
Yields of unthinned loblolly pine
1979.
plantations on cutover sites in the West
Gulf Region. USDA For. Serv. Res. Pap. SO148, 88 p.
South. For. Exp. Sta., New
Orleans, LA
Lohrey, R. E.
Stem volume predictions and crown
1983.
characteristics of thinned longleaf pine
plantations. In:
Proc. 2nd. Biennial
South. Silv. Res. Conf., Atlanta, GA, Nov.
4-5, 1982, p. 338-343. USDA For.Serv. Gen.
Tech. Rpt. SE-24, 514 p.
Southeast. For.
Exp. Sta., Asheville, NC
LITERATURE CITED
Baldwin, V. C. and B. H. Polmer.
1981. Taper functions for unthinned longleaf
pine plantations on cutover West Gulf sites.
Proc. 1st Biennial South. Silv. Res.
In:
Conf., Atlanta, GA, Nov. 6-7,1980, p. 1561 6 3 . USDA For. Serv. Gen. Tech. Rpt. SO-34,
375 P. South. For. Exp. Sta., New Orleans,
LA
SAS Institute.
1979. SAS user's guide.
494 p. Cary, NC
Bennett, F. A., F. T. Lloyd, B. F. Swindel and E.
W. Whitehorne.
Yields if veneer and associated products
1978.
from
unthinned, old-field plantations of
slash pine in the North Florida and South
Georgia
flatwoods.
USDA For. Serv. Res.
Southeast. For. Exp.
Pap. SE-1761 80 p.
Sta., Asheville, NC
SAS Institute, Inc.,
Spurr, S. S.
1952. Forest inventory. The Ronald Press Co.,
xii+ 476 p. New York, NY
Burton, S. S.
The influence of crown ratio on taper
1980.
prediction of loblolly pine (Pinus taeda
M. S. Thesis, vi + 63 py
Virginia
L.).
Poly. Inst. and St. Univ., Blacksburg.
Dell, T. R.
1979..
Potential for using crown ratio in
predicting product yield. In:
Forest
Resource Inventories Workshop Proc., Vol.
II, Ft. Collins, CO, July 1979, p. 843-851.
Dept. For. and Wood Sci., Colorado St.
Univ., Ft. Collins, CO.
Dell, T. R., D. P. Feduccia, T. E. Campbell, W.
F. Mann, Jr., and B. H. Polmer.
1979. Yields of unthinned slash pine plantations on cutover sites in the West Gulf
USDA For. Serv. Res. Pap. SO-147,
Region.
South. For. Exp. Sta., New Orleans,
84 p.
LA
Farrar, R. M., Jr.
Cubic-foot volume, surface area, and
1981.
merchantable height functions for longleaf
pine trees. USDA For. Serv. Res. Pap. SOSouth. For. Exp. Sta., New
166, 7 p.
Orleans, LA
Farrar, R. M., Jr.
1984. A stem-profile function for predicting
multiple-product volumes in natural longleaf
pine trees. USDA For. Serv. Res. Pap. SOSouth. For. Exp. Sta., New
(in process).
Orleans, LA
435