The Crown Ratio and Relative Spacing Relationships for Loblolly Pine Plantation Stands
Plantation Management Research Cooperative
Warnell School of Forestry and Natural Resources University of Georgia
PMRC Technical Report 2009 – 2
December 2009
Dehai Zhao
Michael Kane
Bruce Borders
ii
EXECUTIVE SUMMARY
The data used in this study come from two designed Culture/Density studies for
loblolly pine (Pinus taeda L.). A total of 40 installations were established in the
southeastern United States during 1995-1998. Each installation contains 12 plots of
loblolly pine planted at six levels of density from 600 to 1800 trees/acre in combination
with two levels of management intensity, operational and intensive.
The relationship between the crown ratio and relative spacing index is shown to
be exceedingly predictable. A model for describing this relationship is developed, and
the effects of planting density, site quality, and management intensity on this relationship
are investigated with a nonlinear mixed-effects modeling approach.
When loblolly pine plantation stands reach the average live crown ratio of 0.40, a
critical point representing a generally acceptable level of tree vigor, their values of
relative spacing index range from 0.12 to 0.21, mainly depending on the initial planting
density.
iii
TABLE OF CONTENTS
EXECUTIVE SUMMARY .................................................................................................. II
LIST OF FIGURES ..........................................................................................................IV
LIST OF TABLES.............................................................................................................V
1 INTRODUCTION ........................................................................................................1
2 MATERIALS AND METHODS...................................................................................3
2.1 Study Description .................................................................................................3
2.2 General Model for Crown Ratio – Relative Spacing Index...................................5
2.3 Statistical Analysis ...............................................................................................7
3 RESULTS AND DISCUSSION...................................................................................8
4 LITERATURE CITED ...............................................................................................18
iv
LIST OF FIGURES
Figure 1. Relationships between mean crown ratio and relative spacing for loblolly
pine in culture/density studies in the Lower Coastal Plain and the
Piedmont/Upper Coastal Plain, by planning density and management
intensity. ....................................................................................................6
Figure 2. Original data and fitted crown ratio – relative spacing index curves for
each plot in the loblolly pine Culture/Density study in the Lower Coastal
Plain region..............................................................................................10
Figure 3. Original data and fitted crown ratio – relative spacing index curves for
each plot in the loblolly pine Culture/Density study in the Piedmont and
Upper Coastal Plain region......................................................................11
Figure 4. Crown ratio and relative spacing relationship for loblolly pine stands with
six levels of initial planting density for site index 80 ft under the intensive
management regime in the Lower Coastal Plain (LCP) and
Piedmont/Upper Coastal Plain (PUCP) regions. . ...................................13
Figure 5. Crown ratio and relative spacing relationship for loblolly pine stands for
four levels of site index and initial planting density of 900 trees per acre
under the intensive management regime in the Lower Coastal Plain (LCP)
and Piedmont/Upper Coastal Plain (PUCP) regions. ..............................14
Figure 6. 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/Upper Coastal Plain (PUCP)
regions for 900 trees/acre planting density and 80 ft site index. .............15
v
LIST OF TABLES
Table 1. Maximum likelihood estimates of the parameter, standard errors, and p-
values from the final mixed-effects crown ratio – relative spacing index
models fitted for the culture/density studies in the Lower Coastal Plain (LCP),
and Piedmont/Upper Coastal Plain (PUCP) regions, respectively. .................9
Table 2. Estimated relative spacing when loblolly pine plantations have a crown ratio
of 0.40 by physiographic region, planting density, management intensity, and
site quality. . ..................................................................................................16
1
1 INTRODUCTION
Crown ratio (or live crown ratio; CR), is defined as the proportion of the total
height of a tree that is covered by live branches. CR is considered to be an indirect
measure of a tree’s photosynthetic capacity and a measure of stand density.
Accordingly, diameter and/or height growth equations in many existing forest growth and
yield models use crown ratio as a predictor variable (Monserud and Sterba, 1996; Leites
at al., 2009). CR is also a common indicator of tree vigor, and used to determine the
timing of and potential response of thinning (Bennett, 1955; Long, 1985; Dyer and
Burkhart, 1987). For the southern pines, it is believed that optimum growth and vigor are
maintained when a tree has a live crown ration of 0.33 or higher. Thinning is most
beneficial for stand growth before the average live crown ratio falls below 0.33 (Demers
at al., 2005).
Relative spacing (RS), also known as Hart or Hart-Becking index, 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 as 43,560 / / DRS N H , where N is the
number of trees per acre and HD is average dominant height (ft). 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 developing thinning
specifications for managed plantations (Wilson, 1946 and 1979). Thinning schedules
can be determined by setting proper upper and lower bounds of relative spacing. Stand
needed to be thinned before the relative spacing reaches the lower bound, and post-
thinning relative spacing should not exceed the upper bound. The distance between the
upper and lower relative spacing bounds determines the potential thinning yield. The
desired upper and lower relative spacing bounds for loblolly pine plantations were set at
0.3 and 0.2, respectively, based on previous research (Zhao et al., 2009).
2
In practice, there are several indicators of stand conditions that can aid in
developing thinning schedules, such as the basal area per acre, crown ratio, relative
spacing index, and stand density index (SDI). According to the finding of Long (1985),
for several coniferous species, a live crown ratio of 0.40 seems to correspond with an
SDI of about 0.50 of the maximum SDI for the species. It is natural to ponder the
question - “Are there any predictable relationship among different thinning criteria?” For
example, is there a predictable relationship between the crown ratio and relative
spacing? Can we infer the crown ratio from the relative spacing, or vice versa?
Previous studies (Harrison and Kane, 2008; Zhao et al., 2008) indicated that both
the crown ratio and relative spacing of loblolly pine plantations are influenced by initial
planting density and management intensity. The intensively-managed stands have
lower crown ratio and relative spacing than operationally-managed stands for the same
initial density; both the crown ratio and relative spacing decline with increasing initial
density. Crown ratio and relative spacing decrease with age. A relative spacing model
for loblolly pine plantations developed by Zhao et al. (2009) indicated that the initial
planting density, site index and management intensity affect the development of relative
spacing over time. Tree crown ratio models have been developed for several species to
estimate crown ratio from tree and (or) stand attributes (Dyer and Burkhart, 1987;
Hasenauer and Monserud, 1996; Temesgen et al., 2005). There is relatively little
research on the relationship between crown ration and relative spacing, in spite of
studies of Kanazawa et al. (1985, 1990). If there is a predictable relationship between
crown ratio and relative spacing, is this relationship affected by factors such as initial
density, site quality, and management intensity?
The objective of the present study is to examine how the crown ratio is related to
relative spacing, using the data from loblolly pine culture and density studies across the
southeastern United States. Such relationships and effects of initial planting density, site
3
quality, and management intensity on them are investigated with a nonlinear mixed-
effects modeling approach.
2 MATERIALS AND METHODS
2.1 Study Description
The data used in this study come from two designed culture and density studies
for loblolly pine installed by the Plantation Management Research Cooperative (PMRC)
of the University of Georgia. The Lower Coastal Plain (LCP) Culture/Density Study was
established in 1995/96, having seventeen installations in Georgia, Florida and South
Carolina across five broad soil groups. The Piedmont and Upper Coastal Plain (PUCP)
Culture/Density study 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 subsequent silvicultural
treatments were designed to represent two levels of management intensity: Operational
and intensive culture. In the Lower Coastal Plain 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 preparation. The intensive cultural treatment plots also received
tip moth control through the first two growing seasons and repeated herbicide
applications to achieve complete vegetation control throughout their rotation. At planting,
500 lbs/acre of 10-10-10 fertilizer was applied on all plots. The operation treatment plots
were fertilized with the equivalent of 200 lbs/acre of N and 25 lbs/acre of P in the spring
of the eighth and twelfth growing seasons. The intensive cultural treatment plots also
4
received 600 lbs/acre of 10-10-10 plus micronutrients and 117 lbs lbs/acre of NH4NO3
in the spring of the third growing season, 117 lbs/acre NH4NO3 in the spring of the
fourth growing season, 300 lbs/acre NH4NO3 in the spring of the sixth growing season,
and 200 lbs/acre of N and 25 lbs/acre of P in the spring of the eighth, tenth, and twelfth
growing seasons.
In the Piedmont and Upper Coastal Plain study, any tillage treatments included in
site preparation were carried out on all treatment plots. Both the operational and
intensive treatments included a broadcast chemical 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 operational
and intensive cultural treatment regimes as the Lower Coastal Plain study were applied.
Within both the intensive and operational treatments, six loblolly pine subplots
with densities of 300, 600, 900, 1200, 1500 and 1800 trees/acre 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 surviving seedling after the first
growing season. For detailed information on these two studies such as soils and
treatments carried out for each management level, refer to Harrison and Kane (2008)
and Zhao et al. (2008, 2009), respectively.
Beginning after the second growing season, biennial measurements of diameters
at breast height (Dbh) for all trees and heights (H) on every other tree were made.
Heights to the base of the live crown were measured on all trees that were measured for
total height. Total heights of unmeasured trees were estimated using a height-diameter
equation, 0 1ln( ) /H b b Dbh , fitted to each plot at each measurement age. A tree was
5
considered a dominant tree if it was in the upper 50% of diameters on the plot. Mean
crown ratio was calculated by plot from trees with height measurements, and relative
spacing index were also calculated by plot.
Base age 25 years site index values were estimated for each installation using
the dominant height of the operational treatment plot with 600 trees/acre 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 74.8 to 102.6 ft for the LCP culture/density study
and from 273.5 to 92.3 ft 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 installations in the PUCP culture/density
study remained after 10 growth seasons. Data from these active installations were used
for the analysis reported here.
2.2 General Model for Crown Ratio – Relative Spacing Index
The relationship between crown ratio (CR) and relative spacing index (RS) of the
two loblolly pine culture/density studies is shown in Figure 1. In general, CR is positively
correlated with RS. At early ages, the RS is larger and CR more closely approaches 1.
With stand development, both the RS and the CR become smaller, approaching 0. That
is, crown ratio must be between 0 and 1. This relationship between the RS and the CR
can be described by the following general equation:
2 2
1/( )CR RS RS (1)
where 1 2 and are parameters to be estimated.
6
0.4
0.6
0.8
0.2 0.4 0.6 0.8 1.0 1.2
IPTPH
OPTPH
IPTPH
0.4
0.6
0.8
OPTPH
0.4
0.6
0.8
IPTPH
OPTPH
IPTPH
0.4
0.6
0.8
OPTPH
0.4
0.6
0.8
IPTPH
OPTPH
IPTPH
0.4
0.6
0.8
0.2 0.4 0.6 0.8 1.0 1.2
OPTPH
Relative Spacing Index
Cro
wn
Ra
tio
0.30.40.50.60.70.80.9
0.5 1.0 1.5
IPTPH
OPTPH
IPTPH
0.30.40.50.60.70.80.9
OPTPH
0.30.40.50.60.70.80.9
IPTPH
OPTPH
IPTPH
0.30.40.50.60.70.80.9
OPTPH
0.30.40.50.60.70.80.9
IPTPH
OPTPH
IPTPH
0.30.40.50.60.70.80.9
0.5 1.0 1.5
OPTPH
Relative Spacing Index
Cro
wn
Ra
tio
Figure 1. Relationships between mean crown ratio and relative spacing for loblolly pine in culture/density studies in the Lower Coastal Plain (top) and the Piedmont/Upper Coastal Plain (bottom), by planning density and management intensity.
7
2.3 Statistical Analysis
From Figure 1, we can see there is high between-plot variability. The general
equation (1) is assumed to be common to all plots, but the parameter estimates may
vary across plots. Let and ij ijCR RS denote crown ratio and relative spacing at occasion j
for the plot i; ij denotes the corresponding residual for ijCR . Thus, the general model is
applied to individual plot as
,2 ,2
,1/( )i iijij i ij ijCR RS RS . (2)
The between-plot variation is accounted in the model by taking the parameters as mixed
effects, with random plot effects{ }ib . In the first step we didn’t consider any covariates
on the mixed effects, assuming
(1),1 0
(2),2 0
(1)2
(2) ~ (0, ), ~ (0, ).
i i
i i
iij e
i
b
b
bN N
b
ψ
(3)
Preliminary analysis of the estimated random effects in this case versus planting
density, site index and management intensity clearly indicated that both parameters
decrease as planting density increases. Both parameters seem to have relationship with
site index and management intensity. Thus, in the second step, effects of planting
density, site index and management intensity were taken as fixed to both
parameters ,1 ,2 and i i . After that, statistical test indicates that it is not necessary to
include random plot effects in the parameter i . So the mixed effects model had the
following structure:
8
,2 ,2
,1
,1 0 1 2 3
(2),2 0 1 2 3
(2) 2 2
/( )
( /100)
( /100)
~ (0, ), ~ (0, ),
i iij
e
ij i ij ij
i
i i
i ij
CR RS RS
S TRT PTPA
S TRT PTPA b
b N N
(4)
where PTPA is initial planning density (trees/acre), S is site index (ft), and TRT indicates
management intensity (TRT = 1 for intensive culture, TRT = 0 for operational culture).
Finally, based on the nature of random effects and structure in (4), the effects of
planting density, site index and management intensity on the crown ratio and relative
spacing relationship were tested in terms of parameters 1 2and with the likelihood ratio
test (LRT). The model fit, model comparison, and tests were performed 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.
3 RESULTS AND DISCUSSION
In the crown ratio – relative spacing models for both the LCP and PUCP regions,
only the parameter 2 should be taken as mixed effects, after both the
parameters 1 2 and include the fixed effects of planting density, site index and
management intensity. Based on the model structure (4), the estimates of parameter 1
were not statistically significantly different from zero in both models for the LCP and
PUCP at a significance level of 0.05, indicating that the effect of site index was not
significant in terms of parameter 1 . The estimate of parameter 3 was statistically
significantly different from zero in the models for the PUCP, but not in the model for the
LCP. That is, in terms of parameter 1 there was significant effect of planting density in
9
the model for the PUCP, but no significant effect in the model for the LCP. In terms of
parameter 2 , all the effects of planting density, site index and management intensity
were significant.
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 plot of augmented predictions indicates
that the final models describe the crown ratio – relative spacing index relationship of
individual loblolly pine plots well (Figures 2 and 3), and the residual plots (not shown) did
not indicate any serious deficiencies in the final models.
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/density studies in the Lower Coastal Plain (LCP), and Piedmont/Upper Coastal Plain (PUCP) regions, respectively.
LCP PUCP Parameter Estimates SE p-value Estimates SE p-value
0 0.04223 0.00225 <0.0001 0.03603 0.00216 <0.0001
2 (TRT) -0.01922 0.00257 <0.0001 -0.01886 0.00202 <0.0001 1
3 (PTPA/100) -0.00038 0.00008 <0.0001
0 2.50091 0.10859 <0.0001 3.43744 0.13016 <0.0001
1 (S) -0.00411 0.00105 0.0001 -0.01372 0.00130 <0.0001
2 (TRT) 0.34832 0.04776 <0.0001 0.48732 0.05814 <0.0001
3 (PTPA/100) -0.02552 0.00151 <0.0001 -0.01940 0.00324 <0.0001
2
2 0.06914 0.06104
e 0.04599 0.05114
10
0.20.40.60.8
1.0
0.2 0.8
6112 7112
0.2 0.8
4112 15112
0.2 0.8
13112 9112
0.2 0.8
10112 1112
0.2 0.8
11112 8112
0.2 0.8
12112 14112
0.2 0.8
16112 3112
0.2 0.8
15115 6115
0.2 0.8
1115
4115 13115 10115 9115 14115 11115 8115 12115 7115 16115 3115 4118 6118 12118 9118 1118
0.20.40.60.8
1.011118
0.20.40.60.8
1.015118 10118 8118 14118 7118 16118 13118 3118 14103 8103 6103 4103 12103 3103 9103 13103 16103
10103 15103 1103 7103 11103 13106 9106 8106 11106 1106 14106 6106 4106 7106 12106 10106
0.20.40.6
0.81.0
31060.20.40.6
0.81.0
15106 16106 6109 4109 9109 1109 8109 15109 13109 10109 11109 12109 14109 3109 16109 7109 6012
14012 8012 13012 1012 7012 15012 3012 11012 12012 10012 4012 9012 16012 6015 1015 11015
0.20.40.6
0.81.0
30150.20.40.6
0.81.0
4015 14015 13015 15015 9015 12015 16015 10015 8015 7015 12018 6018 13018 4018 1018 11018 3018
15018 14018 7018 16018 9018 8018 10018 8003 7003 14003 9003 16003 11003 13003 12003 6003
0.20.4
0.60.81.0
30030.20.4
0.60.81.0
15003 10003 1003 4003 6006 14006 7006 8006 9006 1006 10006 11006 15006 13006 3006 4006 12006
16006
0.2 0.8
6009 14009
0.2 0.8
4009 13009
0.2 0.8
1009 9009
0.2 0.8
16009 8009
0.2 0.8
11009 12009
0.2 0.8
15009 10009
0.2 0.8
7009
0.20.4
0.60.81.0
3009
Relative Spacing Index
Cro
wn
Rat
io
Figure 2. Original data and fitted crown ratio – relative spacing index curves for each plot in the loblolly pine Culture/Density study in the Lower Coastal Plain region.
11
0.2
0.6
1.0
0.5 1.5
4112 10112
0.5 1.5
7112 20112
0.5 1.5
3112 9112
0.5 1.5
2112 22112
0.5 1.5
5112 19112
0.5 1.5
23112 17112
0.5 1.5
12112 11112
0.5 1.5
24112 21112
0.5 1.5
8112 1112
0.5 1.5
6112 15112
0.5 1.5
16112 18112
13112 10115 4115 20115 7115 5115 17115 9115 22115 19115 3115 2115 21115 23115 12115 18115 1115 6115 8115 24115 11115
0.2
0.6
1.016115
0.2
0.6
1.013115 15115 5118 7118 10118 20118 17118 4118 21118 2118 22118 12118 23118 24118 3118 9118 19118 18118 11118 1118 6118 8118
16118 15118 13118 16103 13103 9103 15103 5103 11103 8103 18103 7103 17103 1103 24103 6103 23103 2103 20103 10103 12103
0.2
0.6
1.03103
0.2
0.6
1.021103 19103 22103 4103 24106 7106 2106 5106 20106 16106 10106 9106 6106 8106 3106 12106 22106 4106 15106 13106 11106 21106
18106 23106 1106 17106 19106 4109 5109 20109 7109 2109 10109 3109 9109 23109 17109 12109 18109 16109 24109 8109 13109
0.2
0.6
1.06109
0.2
0.6
1.019109 15109 11109 1109 22109 21109 20012 4012 10012 3012 23012 2012 9012 18012 5012 7012 12012 22012 8012 21012 16012 24012
6012 17012 13012 19012 15012 1012 11012 20015 4015 21015 17015 23015 10015 22015 5015 12015 2015 3015 9015 19015 7015
0.2
0.6
1.018015
0.2
0.6
1.08015 6015 16015 11015 24015 13015 1015 15015 4018 20018 7018 21018 22018 3018 10018 9018 12018 24018 5018 2018 23018 18018
19018 17018 11018 8018 6018 16018 1018 13018 15018 8003 18003 9003 23003 20003 4003 3003 2003 17003 10003 15003 13003
0.2
0.6
1.016003
0.2
0.6
1.06003 19003 7003 24003 1003 21003 11003 22003 5003 12003 4006 20006 9006 7006 3006 6006 5006 17006 8006 24006 16006 23006
10006 13006 2006 12006 22006 1006 18006 19006 15006 21006 11006 20009 4009 17009 12009 7009 9009 18009 2009 23009 3009
0.2
0.6
1.010009
0.2
0.6
1.024009
0.5 1.5
5009 21009
0.5 1.5
16009 6009
0.5 1.5
22009 13009
0.5 1.5
8009 1009
0.5 1.5
19009 11009
0.5 1.5
15009
Relative Spacing Index
Cro
wn
Rat
io
Figure 3. Original data and fitted crown ratio – relative spacing index curves for each plot in the loblolly pine Culture/Density study in the Piedmont and Upper Coastal Plain region.
The crown ratio and relative spacing relationship for loblolly pine plantations was
described by
12
2 2ˆ ˆ
1
1
2
ˆ/( )
ˆ 0.04223 0.01922
ˆ 2.5009 0.00411 0.34832 0.02552 ( /100)
CR RS RS
TRT
S TRT PTPA
(5)
for the LCP region; and
2 2ˆ ˆ
1
1
2
ˆ/( )
ˆ 0.03603 0.01886 0.00038 ( /100)
ˆ 3.43744 0.01372 0.48732 0.01940 ( /100)
CR RS RS
TRT PTPA
S TRT PTPA
(6)
for the PUCP regions.
Figure 4 indicates strong effect of initial planting density on the crown ratio and
relative spacing relationship for loblolly pine plantations in both the LCP and PUCP. In
general, stands planted at higher density will have a larger crown ratio than stands with
lower initial planting density when both reach a specific value of relative spacing index.
For example, for loblolly pine stands on sites of 80 ft site index under the intensive
management regime and with planting densities from 300 to 1800 trees/acre, when their
relative spacing decreases to 0.3 the corresponding values of crown ratio ranges from
0.69 to 0.78 in the LCP and from 0.70 – 0.84 in the PUCP; when the relative spacing
decreases to 0.2, then the range of crown ratio will be 0.46 – 0.61 in the LCP, and 0.43 –
0.66 in the PUCP.
In both models (5 and 6), the coefficient of site index was significant and included
in the parameter 2 . For a given initial planting density and management intensity,
however, the effect of site index on the crown ratio and relative spacing relationship is
much smaller in the LCP than in the PUCP (Figure 5).
In terms of both parameters 1 2and in the crown ratio and relative spacing
relationship model, the effect of management intensity was significant for both the LCP
13
and PUCP. The pattern of the crown ratio and relative spacing relationship indicates
that for given site index and initial planting density this relationship of loblolly pine stands
is affected by management intensity mainly before the relative spacing index of these
stands decrease to 0.3. As relative spacing index continues to decrease, the
relationship between the crown ratio and relative spacing is less affected by
management intensity (Figure 6).
Figure 4. Crown ratio and relative spacing relationship for loblolly pine stands with six levels of initial planting density (trees/acre) for site index 80 ft under the intensive management regime in the Lower Coastal Plain (LCP) and Piedmont/Upper Coastal Plain (PUCP) regions. Vertical lines indicate upper (0.3) and lower (0.2) bounds of relative spacing suggested for thinning targets and triggers, respectively.
14
Figure 5. Crown ratio and relative spacing relationship for loblolly pine stands for four levels of site index and initial planting density of 900 trees per acre under the intensive management regime in the Lower Coastal Plain (LCP) and Piedmont/Upper Coastal Plain (PUCP) regions.
15
Figure 6. 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/Upper Coastal Plain (PUCP) regions for 900 trees/acre planting density and 80 ft site index.
16
Given the average crown ratio, the relative spacing can be calculated with the
model after simple algebraic manipulation: 2̂1/1̂[ /(1 )]RS CR CR . A crown ratio of at
least 0.4 has been cited as presenting a generally acceptable level of tree vigor for
number conifers (Long, 1985). When loblolly pine plantations reach average crown ratio
of 0.40, our models suggest that their value of relative spacing will range from 0.12 to
0.21, mainly depending on initial planting density (Table 2). For the southern pines, it is
believed that optimum growth and vigor are maintained before the average live crown
ratio falls below 0.33 (Demers at al., 2005). For this point, the corresponding relative
spacing ranges from 0.10 to 0.19.
Table 2. Estimated relative spacing when loblolly pine plantations have a crown ratio of 0.40 by physiographic region, planting density, management intensity, and site quality. .
Operational Intensive Region
Planting Density SI = 70 SI = 80 SI = 90 SI = 70 SI = 80 SI = 90
300 0.19 0.18 0.18 0.19 0.18 0.18600 0.18 0.17 0.16 0.18 0.17 0.17900 0.17 0.16 0.15 0.17 0.16 0.16
1200 0.15 0.15 0.14 0.16 0.15 0.151500 0.14 0.14 0.13 0.15 0.14 0.14
Piedmont/ Upper Coastal Plain
1800 0.13 0.12 0.12 0.14 0.13 0.13300 0.21 0.19 0.17 0.21 0.19 0.18600 0.20 0.18 0.16 0.20 0.18 0.17900 0.19 0.17 0.15 0.19 0.17 0.15
1200 0.18 0.16 0.14 0.17 0.16 0.141500 0.17 0.15 0.13 0.16 0.16 0.13
Lower Coastal Plain
1800 0.16 0.14 0.12 0.15 0.13 0.12
Kanazawa et al. (1985, 1990) modeled the relationship between the live crown
ratio and relative spacing index for Cryptomeria japonica and the relationship between
relative maximum canopy depth and relative spacing index for Pinus thunbergii in Japan.
17
The formula in their studies can be derived from the nonlinear model (1) with simple
algebraic manipulation. After further logarithmic transformation, the parameters were
estimated by the least squares method in the linear regression. In our study, the general
relationship between crown ratio and relative spacing index was expressed by the
nonlinear model (1), and the parameters were estimated with a nonlinear mixed-effects
modeling approach to address the nature of repeated measures. More importantly, we
concluded that initial planning density, site quality, management intensity, and
physiographic region affect the relationship between crown ratio and relative spacing
index for loblolly pine plantations.
18
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