Open Journal of Forestry
2012. Vol.2, No.3, 110-115
Published Online July 2012 in SciRes (
Copyright © 2012 SciRes.
Crown Ratio and Relative Spacing Relationships
for Loblolly Pine Plantations
Dehai Zhao, Michael Kane, Bruce E. Borders
Warnell School of Forestry and Natural Resources, University of Georgia, Athens, USA
Received April 13th, 2012; revised May 21st, 2012; accepted June 6th, 2012
Two loblolly pine (Pinus taeda L.) culture/density studies were established in 1995-1998 across the
Lower Coastal Plain and Upper Coastal Plain/Piedmont regions of the southern USA. Each installation
contains 12 plots of loblolly pine planted at six levels of density from 741 to 4448 trees/ha in combination
with two levels of cultural intensity, operational and intensive. The data from 37 viable installations were
used to evaluate the crown ratio and relative spacing relationship of loblolly pine plantations. The effects
of planting density, site quality, and cultural intensity on the relationship were investigated with a nonlin-
ear mixed-effects modeling approach. The crown ratio and relative spacing relationship is exceedingly
predictable. When loblolly pine plantation stands reached the average live crown ratio of 0.40, a critical
point representing a generally acceptable level of tree vigor, the corresponding relative spacing index
ranged from 0.11 to 0.20, mainly depending on initial planting density. The information about the crown
ratio and relative spacing relationship would be useful for selecting the best intensity and timing of thin-
Keywords: Loblolly Pine; Intensive Culture; Planting Density; Live Crown Ratio; Relative Spacing Index
Thinning as a forest management practice is employed in
pine plantations for various reasons. For examples, the appro-
priate thinning regimes can reduce density-dependent mortality
rates, increase individual tree growth rates, improve product
assortme nt ratios, a nd enhance spatial and structural uniformity
(Newton, 2009). Practically, there are several indicators of
stand conditions that can aid in determining thinning regimes,
such as live crown ratio (CR), relative spacing index (RS), and
relative stand density (RD). CR is defined as the height of the
live crown (the part of the tree with live branches) divided by
the total height of the tree. CR is a common indicator of tree
vigor (Smith, 1988) and used to determine the timing of and
potential response of thinning (Bennett, 1955; Dyer & Burkhart,
1987; Long, 1985). Tree vigor and normal rate of diameter
growth are maintained as long as CR is 0.40 or greater (Har-
rington, 2000; Smith, 1988), and ideally a thinning treatment
should be scheduled soon after average CR drops below 0.50
(Harrington, 2000). CR, also as an indirect measure of a tree’s
photosynthetic capacity and a measure of stand density, is used
as a predictor variable in many existing forest growth and yield
models (Leites et al., 2009; Monserud & Sterba, 1996).
RS is defined as the ratio of the average distance between
trees to the average dominant height of stand. With square
spacing the ratio is described as10000
SHND, where N
is the number of trees per hectare and HD is average dominant
height (m). RS includes the number of trees and incorporates
both site quality and age through dominant height; thus, it has
been proposed as a useful measure of stand density for devel-
oping thinning specifications for managed plantations (Wilson,
1946, 1979). Thinning schedules can be determined by setting
proper upper and lower bounds of RS. The desired upper and
lower RS bounds for loblolly pine (Pinus taeda L.) plantations
were proposed at 0.3 and 0.2, respectively (Zhao et al., 2010).
RD is the ratio between stand density index (SDI) and maxi-
mum SDI. For loblolly pine the maximum SDI is 450. Stands
begin to undergo density-related mortality (self-thinning) at
0.50 - 0.55 of maximum SDI (Dean & Baldwin, 1993; Drew &
Flewelling, 1979). Therefore, thinning should be scheduled
when RD reaches 0.45 (Dean & Baldwin, 1993; Harrington,
2000). Because these three criteria (CR, RS and RD) could be
considered for selecting the best intensity and timing of thin-
ning, there should be some predictable relationships among
In general, both CR and RS decline over time, but they fol-
low different patterns. CR follows an inverse sigmoidal curve.
In early years, the CR remains very high, with live branches
being retained over nearly 100 percent of stem. As the stand
enters the period of total height rapid development the CR de-
creases rapidly. Then as the stand grows older and the height
rate is slowing down, the CR gradually levels at a minimum
value. The change of RS over time follows a typical inverse-J
trend, which is dependent on the relationship between height
increment and mortality (Parker, 1978). In early years, the RS
changes are due primarily to height growth. With crown closure
the increasing mortality rate plays a more important role, and
RS changes are slowing down. Then RS remains constant with a
minimum value. For RD, it increases and approaches toward 1
as the stand develops, following a sigmoidal curve. There is a
negative relationship between average CR and RD (Long, 1985).
Dean (1999) found that average CR decreases linearly with
increases in RD for loblolly pine plantations in the West Gulf.
He developed a linear model for the CR and RD relationship
and used to manage quality objectives. For the CR and RS,
there is relatively little research on their relationship, except
for some work of Kanazawa et al. (1985, 1990).
Previous studies (Harrison & Kane, 2008; Zhao et al., 2008)
indicated that the CR declining trend over time is significantly
influenced by initial density and management intensity. The CR
decrease rate increases with increasing initial density and man-
agement intensity. Tree CR models for estimating CR from tree
and/or stand attributes have been developed for several species
(Dyer & Burkhart, 1987; Hasenauer & Monserud, 1996; Te-
mesgen et al., 2005), but there is no model developed for di-
rectly describing the CR change through time. The RS devel-
opment through time has been modeled for loblolly pine planta-
tions in the southern United States (Zhao et al., 2010). The
resulting models indicated significant effects of initial density,
site index and management intensity on the RS trend. The in-
tensively managed plots have lower CR and RS than operation-
ally managed plots planted at the same density; with the same
management intensity, both the CR and RS decline with in-
creasing initial density (Zhao et al., 2009, 2010). Therefore, if
there is a predictable relationship between CR and RS, this rela-
tionship is expected to be influenced by initial density, site
index and management intensity.
With data from two loblolly pine culture and density studies
across the southeastern United States, the objective of the pre-
sent study is to test the hypothesis that there is a predictable
relationship between the CR and RS. Moreover, the effects of
initial planting density, site quality, and management intensity
on this relationship are investigated with a nonlinear mixed-
effects modeling approach.
Materials and Methods
Study Description
The data came from two well-designed loblolly pine culture
and density studies initiated by the Plantation Management Re-
search Cooperative (PMRC) of the University of Georgia. The
Lower Coastal Plain (LCP) Culture/Density Study was estab-
lished in 1995/96, with seventeen installations in Georgia,
Florida and South Carolina across five broad soil groups. The
Piedmont and Upper Coastal Plain (PUCP) Culture/Density stu-
dy was established in 1997/98, with twenty-three installations
in Georgia, Alabama, Florida, Mississippi and South Carolina,
stratified over seven broad soil classes.
In both Culture/Density studies, site preparation and subse-
quent silvicultural treatments were designed to represent two
levels of management intensity: operational and intensive cul-
ture. In the LCP study, the operational treatment consisted of
bedding in the spring followed by a fall banded chemical site
preparation; the intensive cultural treatment included bedding
in the spring followed by a fall broadcast chemical site prepara-
tion. The intensive cultural treatment plots also received tip
moth control through the first two growing seasons and re-
peated herbicide applications to achieve complete vegetation
control throughout their rotation. At planting, 561 kg/ha of 10-
10-10 fertilizer was applied on all plots. The operation treat-
ment plots were fertilized with the equivalent of 224 kg/ha of N
and 28 kg/ha of P in the spring of the eighth and twelfth grow-
ing seasons. The intensive cultural treatment plots also received
673 kg/ha of 10-10-10 plus micronutrients and 131 kg/ha of
NH4NO3 in the spring of the third growing season, 131 kg/ha
NH4NO3 in the spring of the fourth growing season, 336 kg/ha
NH4NO3 in the spring of the sixth growing season, and 224
kg/ha of N and 28 kg/ha of P in the spring of the eighth, tenth,
and twelfth growing seasons.
In the PUCP study, any tillage treatments included in site
preparation were carried out on all treatment plots. Both the
operational and intensive treatments included a broadcast chem-
ical site preparation. The operational treatment included a first-
year banded weed control. The intensive cultural treatment
plots received additional herbicide treatments to keep them as
completely free of competing vegetation as possible throughout
their rotation and received tip moth control through the first two
growing seasons. The same fertilizer treatments in the opera-
tional and intensive cultural treatment regimes as the LCP study
were applied.
Within both the intensive and operational treatments, six lob-
lolly pine subplots with densities of 741, 1483, 2224, 2965,
3706 and 4448 trees/ha were randomly located and established
in each installation. To ensure the targeted initial density, each
planting spot was double-planted and reduced to a single sur-
viving seedling after the first growing season. For detailed in-
formation on these two studies such as soils and treatments
carried out for each management level, please refer to Harrison
and Kane (2008) and Zhao et al. (2008, 2010).
Beginning after the second growing season, biennial meas-
urements of diameters at breast height (DBH) for all trees and
heights (H) on every other tree were made. Heights to the base o f
the live crown were measured on all trees that were measured
for total height. Total heights of unmeasured trees were esti-
mated using a height-diameter equation,
ln HbbDBH ,
fitted to each plot at each measurement age. A tree was consid-
ered a dominant tree if it was in the upper 50percent of diame-
ters on the plot. Mean live crown ratio was calculated by plot
from trees with height measurements, and relative spacing in-
dex was also calculated by plot.
Base age 25 years site index values were estimated for each
installation using the dominant height of the operational treat-
ment plot with 1483 trees/ha planting density at the age of the
most recent measurement. Site index was calculated using the
site index equations developed by Borders et al. (2004) for
second rotation loblolly pine plantations. Site indices ranged
from 22.8 to 31.3 m for the LCP Culture/Density Study and
from 22.4 to 28.1 m for the PUCP Culture/Density Study.
After 12 growth seasons 14 of the original 17 installations in
the LCP culture/density study were viable; and all 23 installa-
tions in the PUCP culture/density study remained after 10
growth seasons. Data from these active installations were used
for the analysis reported.
Model Development
Plot examination of the CR versus RS indicated that CR is
positively correlated with RS. While CR more closely approaches
1 in early ages, the RS is larger. With stand development both the
RS and CR become smaller, approaching to their different mini-
mum values. To constrain CR predictions between 0 and 1, the
following general equation was used to describe the relationship
between the CR and RS:
where 12
are the paramet ers.
Preliminary analysis indicated that both parameters 1
varied across installations and plots. These between-in-
stallation and between-plot variations may be accounted in the
Copyright © 2012 SciRes. 111
Copyright © 2012 SciRes.
model by taking the parameters as mixed effects. Further
evaluation that the parameters 1
and 2 had some linear
relationships with initial planting density (N0, trees/ha), man-
agement intensity (TRT, a dummy variable indicating the man-
agement intensity: TRT = 1 for intensive culture and TRT = 0
for operational culture), and/or site index (SI, m). Thus, the ef-
fects of planting density, site index and management intensity
were taken as fixed to both parameters 1 and 2. After in-
troducing fixed-effects of planting density, management inten-
sity, especially of site index into two parameters, only the plot
effects are considered random, denoted as
 
b, and included
in the parameters. Let ij and ij
S denote crown ratio and
relative spacing at occasion j for the plot i; ij denotes the
corresponding residual forij . Thus, the mixed effects model
is applied to individual plot as
parameter 2
was taken as mixed effects. The random-effects
variance-covariance matrixwas changed to that is, ψ22,
(2) 2
0, SI was not a significant predictor of the
parameter 1
for either of the studies at . Planting
density was significant at 0.05
in terms of parameter 1
for the PUCP Culture/Density study, but no significant for the
LCP Culture/Density study. In terms of parameter 2
, all the
effects of planting density, site index and management intensity
were significant at 0.05
After excluding non-significant covariates from the model,
we determined and refitted the final model. Parameter estimates
of the final model for each of the two culture/density studies are
given in Table 1. The CR-RS relationship for loblolly pine
plantations was described by
,10 10
,20 10
(1) 2
~ 0,.
NN ~
ˆ0.04223 0.01922
ˆ2.50108 0.013470.34832
0.01033 100
 
 
(2) for the LCP region; and
ˆ0.03602 0.018860.00015100
ˆ3.43792 0.04503
 
The random-effects terms and their corresponding appropri-
ate variance-covariance matrixψin (2) are further identified.
Then, based on the final nature of random effects and structure
in (2), the effects of planting density, site index and manage-
ment intensity on the CR and RS relationship were tested in
terms of parameters 1 and 2 with the likelihood ratio test
(LRT). The model fit, model comparison, and tests were per-
formed using the NLME library by Pinheiro and Bates (2000)
for S-plus software, and separately for the data from the LCP
and PUCP Culture/Density studies.
for the PUCP regions. The augmented prediction plots indi-
cated that the final models describe the CR-RS relationship of
individual loblolly pine plots well, and the residual plots did not
indicate any serious deficiencies in the final models.
As stands develop, both the CR and RS decrease. Although
the changes of CR and RS over time follow different patterns—
an inverse sigmoidal curve and a typical inverse-J curve, re-
spectively, the positive relationship between CR and RS can
still be described by Model (1). The resulting CR-RS models for
loblolly pine plantations in the LCP and PUCP regions, Models
(3) and (4), indicated that the relationship between CR and RS
is exceedingly predictable.
Results and Discussion
In the CR-RS models for both the LCP and PUCP regions,
after including the fixed-effects of planting density, site index
and management intensity into both the parameters 1
Our results showed that initial planting density significantly
affected the CR and RS relationship for both regions (Figure 1).
In general, stands planted at higher density will have a larger
CR than stands planted at lower density when both reach a
and 2
the plot random-effects was no significant on the parameter 1
but still significant on the parameter . Therefore, only the
Table 1.
Maximum likelihood estimates of the parameter, standard errors (SE), and p-values from the final mixed-effects crown ratio—relative spacing index
models fitted for the culture/ d e nsity studies in the Lower Coastal Plain (LCP), and Piedmont/Upper Coastal Plain (PUCP) r e g i o n s , respectively.
Parameter Estimates SE p-value Estimates SE p-value
0.04223 0.00225 <0.0001 0.03602 0.00216 <0.0001
(TRT) –0.01922 0.00257 <0.0001 –0.01886 0.00202 <0.0001
(N0/100) –0.00015 0.00003 <0.0001
2.50108 0.10578 <0.0001 3.43792 0.13017 <0.0001
(S) –0.01347 0.00345 0.0001 –0.04503 0.00425 <0.0001
(TRT) 0.34832 0.04776 <0.0001 0.48737 0.05813 <0.0001
(N0/100) –0.01033 0.00061 <0.0001 –0.00786 0.00131 <0.0001
0.06913 0.06106
0.04599 0.05113
specific RS. For loblolly pine stands on site index of 24 m and
with the intensive management regime, when their RS de-
creases to 0.3, as planting density increases from 741 to 4448
trees/ha, the CR increases from 0.69 to 0.78 in the LCP and
from 0.70 to 0.84 in the PUCP, respectively. When the RS
reaches 0.2 the CR ranges from 0.46 to 0.61 in the LCP, and
from 0.43 to 0.66 in the PUCP, respectively.
In terms of parameter in the CR-RS model, site index was
significant for both regions. For a given planting density and
management intensity, however, the effect of site index on
overall CR-RS relationship is much smaller in the LCP than in
the PUCP (Figure 2). In terms of both parameters 1
and 2
the effect of management intensity was significant for both the
LCP and PUCP. However, further examination of overall CR-
RS relationship suggested that, for a given site index and initial
density, management intensity strongly affect loblolly pine
Figure 1.
Crown ratio and relative spacing relationship for loblolly pine stands
with six levels of initial planting density (trees/ha) for site index 24
m under the intensive management regime in the Lower Coastal
Plain (LCP) and Piedmont/Upper Coastal Plain ( PUCP) regions.
Figure 2.
Crown ratio and relative spacing relationship for loblolly pine stands
for four levels of s ite index and in itial planting density of 2224 trees/
ha under the intensive management regime in the Lower Coastal
Plain (LCP) and Piedmont/Upper Coastal Plain ( PUCP) regions.
CR-RS relationship before the RS decreases to 0.2 or before the
CR reaches 0.5. During that period, intensively-managed plots
will have larger CR than operational plots when they reach the
same value of RS. After that period, the CR-RS relationship is
less affected by management intensity (Figure 3 and Table 2).
Given the average CR, the RS can be calculated with the
model after simple algebraic manipulation, i.e.,
 
When loblolly pine plantations reach average CR of 0.40, a
generally acceptable level of tree vigor for numerous conifers
(Long 1985; Smith 1988), our models suggested that the RS
value will range from 0.11 to 0.20, mainly depending on initial
planting density (Table 2). For southern pines, Demers et al.
(2005) believed that optimum growth and vigor are maintained
before the average CR falls below 0.33. Based on our models,
the RS corresponding to 0.33 CR for loblolly pines will range
from 0.10 to 0.18. Harrington (2000) suggested that ideally a
thinning treatment should be scheduled soon after average CR
drops below 0.50 for loblolly pine. The RS corresponding to
0.50 CR for loblolly pines ranges from 0.14 to 0.25.
According to Dean’s model for the CR-RD relationship for
loblolly pines in the West Gulf Dean (Dean, 1999), the RD cor-
responding to 0.40 average CR is 0.63. Usi ng data from our lob-
lolly pine culture and density studies, the negative CR-RD rela-
tionship is described by the equation: .
Furthermore, this relationship is not influenced by planting
density, site index, and management intensity. Based on this
model, corresponding to average CR of 0.4 for loblolly pines in
the southern US the RD is 0.87, and corresponding to average
CR of 0.5 the RD is 0.72. It was suggested that thinning should
be scheduled for loblolly pine plantations when the RD reaches
0.45 (Dean & Baldwin 1993; Harrington, 2000), which corre-
sponds to 0.68 CR. It is obvious that simultaneously consider-
ing the CR and RD as thinning triggers may result in different
decision on timing of thinning.
0.9751 0.6623CR RD
Figure 3.
Crown ratio and relative spacing relationship for loblolly pine
stands for two levels of management intensity (operational and
intensive) in the Lower Coastal Plain (LCP) and Piedmont/Up-
per Coastal Plain (PUCP) regions for 2224 trees/ha planting
density and 24 m site index.
Copyright © 2012 SciRes. 113
Table 2.
Estimated relative spacing when loblolly pine plantations have a crown ratio of 0.40 by physiographic region, planting density, management intensity,
ality. and site qu
Operational Intensive
Region Planting Density (trees/ha) SI = 22SI SI = 28SI = 22 SI = 26SI = 28= 24SI = 26 24 SI =
741 0.19 0.18 0.18 0.17 0.19 0.18 0.18 0.17
Piedmont/Upper Coastal Plain
85 0.18 0.17 0.17 0.16 0.18 0.17 0.17 0.17
2224 0.16 0.16 0.16 0.15 0.17 0.16 0.16 0.16
2965 0.15 0.15 0.14 0.14 0.16 0.15 0.15 0.15
3706 0.14 0.14 0.13 0.13 0.15 0.14 0.14 0.14
r Coastal
4448 0.13 0.13 0.12 0.12 0.14 0.13 0.13 0.13
741 0.20 0.19 0.18 0.17 0.20 0.19 0.18 0.17
85 0.19 0.18 0.17 0.16 0.19 0.18 0.17 0.16
2224 0.18 0.17 0.16 0.15 0.18 0.17 0.16 0.15
2965 0.17 0.16 0.15 0.13 0.17 0.16 0.15 0.14
3706 0.16 0.15 0.14 0.12 0.15 0.14 0.13 0.12
4448 0.15 0.14 0.13 0.11 0.14 0.13 0.12 0.11
When calculating the RD for loblolls, the maxi
DI is accepted as 450. However, the SDI of some plots in our
lolly pine planta-
This study wastion Management
Research Cooperasity of Georgia’s
Bennett, F. (1955). Thee height and diameter
growth of planted slary, 53, 636-638.
y pinemum
Slture and density studies was actually greater than 450.
Therefore, when the RD is considered as one of thinning crite-
ria, choosing the maximum SDI is important.
In summary, there is a predictable relationship between live
crown ratio and relative spacing index for lob
ns. This relationship is affected by initial planning density,
site quality, management intensity, and physiographic region.
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