Relation of Initial Spacing and Relative Stand 

Density Indices to Stand Characteristics in a 

Douglas-fir Plantation Spacing Trial 

Robert O. Curtis, Sheel Bansal, and Constance A. Harrington

United States Department of Agriculture

Forest 

Service

Pacific Northwest 

Research Station

Research Paper

PNW-RP-607 

April

2016
Authors

Robert O. Curtis is a scientist emeritus, Sheel Bansal was a research ecologist, 

and Constance A. Harrington is a research forester, U.S. Department of Agriculture, Forest Service, Forestry Sciences Laboratory, 3625 93rd Ave. SW, Olympia, 

WA 98512-9193. Bansal is currently a research ecologist, U.S. Geological Survey, 

8711 37th St. SE, Jamestown, ND 58401.

Cover: Young Douglas-fir stand next to older stand. Photo by U.S. Forest Service 

Pacific Northwest Region.

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Abstract

Curtis, Robert O.; Bansal, Sheel; Harrington, Constance A. 2016. Relation 

of initial spacing and relative stand density indices to stand characteristics in 

a Douglas-fir plantation spacing trial. Res. Pap. PNW-RP-607. Portland, OR: 

U.S. Department of Agriculture, Forest Service, Pacific Northwest Research 

Station. 27 p.

This report presents updated information on a 1981 Douglas-fir (Pseudotsuga 

menziesii (Mirb.) Franco var. menziesii ) plantation spacing trial at 33 years from 

planting. Stand statistics at the most recent measurement were compared for initial spacing of 1 through 6 meters and associated relative densities. There was no 

clear relationship of spacing to top height. Diameter, live crown ratio, and percent 

survival increased with spacing; basal area and relative density decreased with 

increase in spacing. Volume in trees ≥ 4 cm diameter was greatest at 2 m spacing, while utilizable volume (trees ≥20 cm dbh) was greatest at 4 m spacing. Live 

crown ratio decreased and total crown projectional area increased with increasing 

relative density indices. Total crown projectional area was more closely related to 

relative density than to basal area. 

Keywords: plantation spacing, stand density, crown dimensions, yield, SDI, 

RD, Pseudotsuga menziesii.
Summary

This report presents updated information on a 1981 Douglas-fir (Pseudotsuga 

menziesii (Mirb.) Franco var. menziesii ) plantation spacing trial at 33 years from 

planting (35 from seed). Initial planting spacings were 1, 2, 3, 4, 5 and 6 m. Stand 

statistics by spacing and associated values of relative density at the most recent 

measurement (2013) were compared. There was no clear relationship of spacing 

to top height. Diameter, live crown ratio, and survival percentage increased with 

spacing; basal area and relative density decreased with increase in spacing. Volume 

in trees ≥ 4 cm diameter was greatest at 2 m spacing, while utilizable volume (trees 

≥20 cm dbh) was greatest at 4 m spacing. Several common relative density indices 

have near-perfect correlations in these data and are essentially equivalent, except for 

proportionality constants. Live crown ratio decreased and total crown projectional 

area increased with increase in relative density. Total estimated crown projectional 

areas suggest that crown closure occurs at about 1/3 of maximum relative density. 

Total crown projectional area was more closely related to relative density than to 

basal area, suggesting that relative density may be a better predictor of shading. 
Contents

1 Introduction

1 The Study

1 Study Area

2 Study Design

2 Methods and Results

2 Height

3 Diameter

6 Number of Trees

8 Slenderness Ratio

8 Basal Area

9 Volume

10 SDIsum

12 Live Crown Ratio

12 Crown Width

14 Crown Area

15 Discussion

15 Relative Stand Density

16 Maximum Relative Density

16 Volume

17 Live Crown Ratio and Height to Live Crown

19 Crown Width and Crown Projectional Area

23 Conclusion

24 English Equivalents

24 References

26 Acknowledgments

26 Glossary of Symbols and Acronyms

27 Appendix

1

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Introduction

This report discusses developments in the “new” Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco var. menziesii) plantation spacing trial at the Wind River Experimental Forest. This trial is called “new” to distinguish it from the pioneering early 

trial (Reukema 1979) established at Wind River in 1925. The 1925 trial showed that 

wider spacing than was then in common use had considerable advantages both in 

reduced costs and superior tree development, a result that had a strong influence on 

planting practices in the Pacific Northwest. But, interpretation of results of the 1925 

trial has been somewhat clouded by defects in the original experimental design 

(Miller et al. 2004), plus the fact that it represents only a single site of rather poor 

quality.

In the late 1950s and following decades, a number of organizations undertook 

additional plantation spacing trials with improved designs and including a wider 

range in geography and site quality.

The Study

We here present a progress report on one of these, the “new” Wind River spacing 

trial, established in 1981. The study differs from most previous trials in that it 

includes a wider range of initial spacings and more extensive replication of spacings 

within a single study area. Our objectives are to (1) update and supplement information given in a previous report by Harrington et al. (2009), and (2) provide some 

additional information on crown development in relation to initial spacing and stand 

density indices.

Study Area

The study area is in the Trout Creek Unit of the Wind River Experimental Forest 

(T4N R7E S18) on the Gifford Pinchot National Forest, near Carson, Washington. 

Elevation is 512 to 557 m, with slopes less than 10 percent and a southerly aspect. 

The soil is classified as a dark brown loam of the Stabler series, a medial, amorphic, mesic, Vitric Hapludand (Harrington et al. 2009). Preharvest vegetation was 

intermediate between the western hemlock (Tsuga heterophylla (Raf) Sarg.) and 

Pacific silver fir (Abies amabalis Dougl. Ex Forbes) zones (Franklin and Dyrness 

1973, Harrington et al. 2009).

The area was clearcut in 1977, slash burned in 1978, and planted in spring 1981 

with 2-0 stock. Herbicide was applied in summer 1981 to reduce vegetative competition with the planted seedlings. Naturally established seedlings were removed in 

1986, 1990, and 1998.

Although the present stand is still too young for a precise site index estimate, it 

now appears to be a high site 3 (King 1966).
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RESEARCH PAPER PNW-RP-607

Study Design 

Planting spacings were 1, 2, 3, 4, 5, and 6 m. Treatment plot size was 0.4 ha, with 

measurements taken on interior subplots of size varying with spacing (table 1).

We consider here only the 38 plots planted with pure Douglas-fir (some additional plots were mixed species). Assignment of treatments was randomized within 

the study area. There were four plots each in the 1 m spacing, ten plots in the 3 m 

spacing, and six plots each in the other spacings. Arrangement of plots is shown in 

figure 1.

Additional and more detailed information has been given by Harrington et al. 

(2009), including results from complete remeasurements in 1997, 2001, and 2005.

Methods and Results

The plots were most recently remeasured in fall 2013, 33 growing seasons after 

planting and 35 years from seed. This report presents results from the 2013 remeasurement, extending information given in the 2009 report. We also provide some 

comparisons of crown development in relation to spacing and relative density 

measures, additional to the information in the 2009 report. 

Results are given in metric units for consistency with Harrington et al. (2009). 

We also give equivalent values in English units in parentheses, where this can be 

done conveniently. Table 7 in the appendix gives spacing means in English units.

Height

Heights and heights to live crown (HLC) were measured on all trees in the 5 m and 

6 m spacings. Heights and HLC in other spacings were measured on a sample of 

trees, usually about 30 to 40 trees per plot distributed to cover the range of diameters, with about two thirds in the upper one-half of the diameter range. Height/

diameter equations were fit to the data for each plot and these were used to supply estimated heights for those trees lacking measured heights. Stand height was 

expressed as top height (H100), defined as the mean height of those Douglas-fir 

included in the largest (by diameter) 100 trees per hectare (40/ac).

Table 1—Plot dimensions and initial number of trees, by spacing

Spacing Measurement plot Number of trees planted

Meters Feet Hectares Acres Per hectare Per acre

1 3.2 0.020 0.049 10,000 4,049

2 6.6 0.080 0.198 2,500 1,012

3 9.8 0.176 0.435 1,111 450

4 13.1 0.160 0.395 625 253

5 16.4 0.160 0.395 400 162

6 19.7 0.176 0.435 278 113
3

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 1—Physical arrangement of plots, with planting spacings.

Mean H100 in 2013 was 22.0 m (72 ft). Figure 2 compares spacing means of 

H100 and HLC in 2013, ± 1 standard error. Although figure 2 suggests that height 

growth may have been slightly reduced at the extremes of spacing, there is no clear 

evidence of an effect of spacing on top height. A regression of form H100 = a + 

b(spacing) + c(1/spacing) fit to the 38 plot values produced estimates in close agreement with the spacing means, but was not statistically significant (p = 0.17); a result 

consistent with Harrington et al. (2009).

Diameter (D)

Diameters were measured on all trees ≥ 4.0 cm diameter at breast height (dbh), 

and are here expressed in terms of two summary values. These are (1) D100, mean 

diameter of the 100 largest trees per hectare, and (2) QMD, the quadratic mean 

diameter of all live trees of diameter ≥ 4.0 cm (1.6 in). Values of D100 increased 

from 23.1 cm (9.1 in) in the 1 m spacing to 34.0 cm (13.4 in) in the 6 m spacing. 

Values of QMD increased from 11.3 cm (4.45 in) in the 1 m spacing to 28.55 cm 

(11.24 in) in the 6 m spacing. Spacing means ± 1 SE are shown in figure 3. D100 

development over time is shown in figure 4, and QMD development over time in 

figure 5.
4

RESEARCH PAPER PNW-RP-607

0

5

10

15

20

25

1 2 3 4 5 6

Mean H100 and

mean H

L

C100 (m)

Spacing (m)

H100

HLC100

Figure 2—Spacing means of H100 and HLC100 at age 33 years from planting, 35 from seed, ± one 

standard error.

Figure 3—Spacing means of D100 and QMD at 33 years from planting, 35 from seed, 

± one standard error.

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7

Diameter (cm)

Spacing (m)

D100

QMD
5

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 4—Time trends of D100 by spacing and age from seed.

Figure 5—Time trends of QMD by spacing and age from seed.

0

5

10

15

20

25

30

35

40

15 20 25 30 35 40

D100 (cm)

Years from seed

1 m

2 m

3 m

4 m

5 m

6 m

0

5

10

15

20

25

30

15 20 25 30 35 40

Quadratic mean diameter (cm)

Years from seed

1 m

2 m

3 m

4 m

5 m

6 m
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RESEARCH PAPER PNW-RP-607

Number of Trees

Figure 6 compares mean numbers of live trees of diameter 4.0 cm and larger at total 

age 35, by spacing. These ranged from 4387/ha (1,897/ac) in the 1 m spacing, to 279/

ha (113/ac) in the 6 m spacing. Figure 7 shows trends over time in number of live 

trees. Figure 8 compares survival as a fraction of the number of trees originally 

planted. As would be expected, mortality has been extreme at the 1 × 1 m spacing, 

and decreased sharply as spacing increased.

Figure 6—Spacing means of number of live trees ≥ 4 cm, at 33 years from planting, 35 from seed, ± 

one standard error.

0

1,000

2,000

3,000

4,000

5,000

6,000

1 2 3 4 5 6

Num

ber/ha

Spacing (m)
7

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 7—Time trends of number of live trees ≥ 4.0 cm, by spacing and age from seed.

Figure 8—Number of live trees ≥ 4 cm at 33 years from planting, 35 from seed, expressed as 

a fraction of number planted. Error bars are ± one standard error.

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

15 20 25 30 35 40

Num

ber per hectare

Years from seed

1 m

2 m

3 m

4 m

5 m

6 m

1.2

1.0

0.8

0.6

0.4

0.2

0

1 2 3 4 5 6

Num

ber of surviv

ors/num

ber planted

Spacing (m)
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RESEARCH PAPER PNW-RP-607

Slenderness Ratio

The ratio Height/Diameter, in which both are in the same units, is related to 

competition and is sometimes used as an indicator of windfirmness and extent of 

crown abrasion (Wilson and Oliver 2000). The ratio decreased with increase in 

spacing as shown below for (1) the 100 largest trees per ha (40 per acre), and (2) 

all trees ≥ 4 cm.

Spacing H100/D100 Hqmd/QMD

Meters Mean ± one standard error

1 90.6 ±2.5 121.9 ± 2.6

2 83.5 ± 2.2 105.7 ± 2.1

3 74.5 ± 0.8 91.0 ± 1.2

4 67.4 ± 1.0 79.0 ± 1.2

5 65.2 ± 0.4 72.8 ± 1.2

6 63.6 ± 0.7 67.4 ± 0.8

Basal Area

Basal area decreased with increased spacing, from 45.6 m2/ha (199 ft2/ac) in the 1 

m spacing, through 19.2 m2/ha (84 ft2/ac) in the 6 m spacing. Means of plot basal 

areas by spacing at total age 35, ± 1 standard error, are shown in figure 9. Time 

trends of basal area, by spacing and age from seed, are shown in figure 10.

Figure 9—Spacing mean basal areas at age 33 from planting, 35 from seed, ± one standard error.

60

50

40

30

20

10

0

1 2 3 4 5 6

Basal area (m2/ha)

Spacing (m)
9

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Volume

Volumes of stem wood in 2013 were calculated using the Bruce and DeMars (1974) 

volume equation. Measured heights were used when available; otherwise, heights 

were estimated using the H/D equations mentioned above. Two cubic volumes per 

hectare were calculated: (1) CV4, cubic volume of entire stem for trees of diameter 

≥ 4.0 cm, and (2) CV20, the corresponding value for trees of diameter ≥ 20.0 cm 

(~8 in). The latter value was an arbitrary choice, as an approximation to trees that 

would likely be commercially utilizable. Values of CV4 at total age 35 peaked at 2 

m spacing and decreased with increase in spacing, ranging from 301 m3/ha (4,306 

ft3/ac) in the 1 m spacing, to 342 m3/ha (4,896 ft2/ac) in the 2 m spacing, to 140 

m3/ha (2,001 ft3/ac) in the 6 m spacing (table 2). Maximum CV20 was 211.9 m3 ha 

(3,030 ft3/ac) in the 4 m spacing.

Figure 11 compares the means of the plot estimates of CV4 and CV20 at total 

age 35, by spacing, with their associated standard errors. Time trends of CV4 are 

shown in figure 12, by spacing and age from seed.

Figure 10—Time trends of spacing mean basal areas, by spacing and age from seed.

0

5

10

15

20

25

30

35

40

45

50

15 20 25 30 35 40

Basal area (m2/ha)

Years from seed

1 m

2 m

3 m

4 m

5 m

6 m
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RESEARCH PAPER PNW-RP-607

SDIsum

This variable, defined as SDIsum = ∑(di/25.4)1.6/area, is the SDI* of Long and 

Daniel (1990) and SDIsum of Curtis (2010), converted to metric units. It is one of 

several expressions of relative stand density that are essentially the same except for 

scale factors.

Spacing mean values of SDIsum ± 1 standard error at total age 35 are shown 

in figure 13. As would be expected, they form a relatively smooth trend of SDIsum 

decreasing with increasing spacing.

Figure 11—Means of cubic meters per hectare of trees with diameters ≥ 4cm (CV4) and ≥ 20 cm 

(CV20) at 33 years from planting, 35 from seed, by spacing. ± one standard error.

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7

Volume (m3/ha)

Spacing (m)

CV4 cm+

CV20 cm+

Table 2—Cubic volumes at 27 and 35 years from seed, in trees ≥ 4 cm

Spacing Spacing 2005 2013 PAIa PAIa

Meters Feet

Cubic meters/

hectare

Cubic meters/

hectare

Cubic meters/

hectare/year

Cubic feet/

acre/year

1 3.28 215.4 301.1 10.71 153

2 6.56 215.1 342.4 15.91 228

3 9.84 148.6 265.1 14.56 286

4 13.12 113.3 228.6 14.41 206

5 16.40 92.4 182.9 11.31 162

6 19.68 62.0 139.9 9.74 139 a Periodic annual increment.
11

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 12—Time trends of cubic volume per hectare of trees with diameters ≥ 4 cm (CV4), by 

spacing and age from seed.

0

50

100

150

200

250

300

350

400

1 m

2 m

3 m

4 m

5 m

6 m

15 20 25 30 35 40

Years from seed

CV4 (m3

/ha)

Figure 13—Spacing means of SDIsum at 33 years from planting, 35 from seed, ± one standard error.

0

200

400

600

800

1000

1200

1400

S

DIsum

1 2 3 4 5 6

Spacing (m)
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RESEARCH PAPER PNW-RP-607

Live Crown Ratio (LCR)

Height to live crown was measured on all trees measured for total height.

Live crown ratio was calculated as: 

LCR = (tree height – height to live crown)/ tree height

Regressions LCR = a + bD + cD2 + d/D were fit, separately for each plot, 

retaining only significant terms. These regressions were then used to estimate two 

values used to characterize each plot, (1) LCR100, the live crown ratio corresponding to the value of D100, and (2) LCRqmd, the live crown ratio corresponding to the 

plot quadratic mean diameter. Figure 14 compares these values at total age 35, by 

spacing. As expected, the difference between LCR100 and LCRqmd decreases as 

spacing increases, and both increase as spacing increases. Figure 15 shows trends 

over time by spacing and age from seed. Figure 16 shows the trend of LCR100 in 

relation to SDIsum at the 2013 measurement.

Crown Width (CW)

Crown widths were measured on a much smaller subsample of trees, frequently 8 to 

12 trees per plot but sometimes much less.

Crown width was calculated as the mean of two crown diameter measurements 

taken at right angles, to the outermost branch tip. Per plot sample sizes were often 

very small, particularly in the 1 m and 2 m spacings. Therefore, one regression of 

crown width on tree diameter was fit to pooled measurements from all plots at a 

given spacing, using the best subset from CW = a + bD + cD2 + d/D. Similar computations were made for the 1997, 2001 and 2005 measurements. Estimated 2013 

crown widths corresponding to the spacing quadratic mean diameter are shown 

in table 3.

Table 3—Estimated crown widths corresponding to quadratic mean diameter 

of trees ≥ 4.0 cm at 35 years from seed

Spacing QMD Estimated CW QMD Estimated CW

Meters Centimeters Meters Inches Feet

1 11.31 2.48 4.45 8.1

2 17.04 3.86 6.71 12.7

3 20.52 4.33 8.09 14.2

4 24.92 5.27 9.81 17.3

5 27.26 5.60 10.73 18.4

6 28.35 5.77 11.16 18.9
13

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 14—Spacing live crown ratios corresponding to (1) D100, and (2) QMD, at 33 years from 

planting, 35 from seed, ± one standard error.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Live cro

wn ratio

1 2 3 4 5 6

Spacing (m)

LCR100

LCRqmd

Figure 15—Time trends of LCR100 over spacing.

0

0.20

0.40

0.60

0.80

1.00

1.20

0 1 2 3 4 5 6 7

L

C

R100

Spacing (m)

19 years from seed

23 years from seed

27 years from seed

35 years from seed
14

RESEARCH PAPER PNW-RP-607

Table 4—Spacing means at age 35 from seed, ± 1 SE

Spacing ∑CA/area SDIsuma RDsuma RDqmda Treesa DIsumb RDsumb RDqmdb Treesb

Meters - - - - - - - - - - - - - Per hectare - - - - - - - - - - - - - - - - - - - - - - - Per acre - - - - - - - - - -

1 2.15 ± 0.10 1191 ± 72 12.89 ± 0.79 13.6 ± 0.78 4588 ± 383 482 89.6 94.3 1857

2 2.10 ± 0.11 973 ± 49 10.10 ± 0.47 10.4 ± 0.50 1900 ± 51 394 70.1 73.2 769

3 1.33 ± 0.05 687 ± 30 7.01 ± 0.30 7.2 ± 0.37 983 ± 27 278 48.7 50.5 398

4 1.18 ± 0.04 543 ± 25 5.45 ± 0.24 5.6 ± 0.25 570 ± 10 220 37.9 38.8 231

5 0.92 ± 0.02 432 ± 24 4.19 ± 0.24 4.4 ± 0.23 393 ± 10 175 29.1 30.6 175

6 0.74 ± 0.05 336 ± 25 3.26 ± 0.24 3.3 ± 0.23 286 ± 9 136 22.7 23.2 136

a Metric units.

b English units

Crown Area (CA)

Crown projectional area was calculated for each tree, as CA = 0.7854 (CW)2, using 

measured CW when available, otherwise estimates from the CW = f(D) regressions. 

Estimated crown areas were summed over all trees on a plot and divided by ground 

area of the plot, to provide plot estimates of ∑CA/ha. Means of the resulting values 

are shown in table 4 for each spacing, together with the corresponding spacing 

means of SDIsum, RDsum, and number of trees ≥ 4 cm per ha.

Figure 16—Trend of LCR100 in relation to SDIsum, at 33 years from planting, 35 from seed, with 

fitted regression curve.

0

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 200 400 600 800 1000 1200 1400 1600

L

C

R100

SDIsum

Observed

LCR100 = a + b × SDIsum + c/SDIsum + d × sqrt(SDIsum)
15

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Table 5— Correlation coefficients (r) among some relative 

density values calculated from the 2013 measurements

SDIqmd RDsum RDqmd

SDIsum 0.9989 0.9987 0.9977

SDIqmd 0.9990 0.9984

RDsum 0.9996

Discussion

Relative Stand Density 

In this report, we have used SDIsum (=∑(di/25.4)1.6/ha) as the measure of relative 

stand density. This is the metric equivalent of the SDI* of Long and Daniel (1990) 

and the SDIsum of Curtis (2010), and was an arbitrary choice among several very 

similar measures.

Two relative density measures in common use in the Douglas-fir region are 

Reineke SDI (Reineke 1933) and Curtis RD (=basal area/(QMD)0.5; Curtis 1982), 

here symbolized as SDIqmd and RDqmd. Their computation involves QMD, which 

is strongly influenced by departures from the more or less symmetrical diameter 

distribution of truly even-aged stands, particularly skewed distributions arising 

from the presence of small trees of a younger age class. There are a number of 

closely related relative density measures that do not involve the calculation of QMD 

and are less influenced by skewed diameter distributions. 

In the following discussion, we use the term “relative density” to denote a class 

of very closely related measures that do not use QMD and are nearly equivalent. 

These include SDIsum (used here), RDsum (Curtis 2010), and Tree-Area-Ratio 

(Curtis 1971). Algebraic manipulation of the basic formulas shows that all three of 

these can be expressed in the form “a[∑dc/area]”, where “a” is a factor specific to 

the relative density measure and measurement system (metric vs. English), and the 

exponent “c” has value 1.6 for SDIsum, 1.5 for RDsum, and 1.55 in Curtis TAR. 

Values of these measures are very highly correlated in these data (table 5). They 

are nearly proportional and aside from proportionality factors, differ only by small 

differences in the exponent of diameter. Any of these measures would have given 

results very similar to those shown here. We have included RDsum and RDqmd 

in table 5 because RDqmd is a widely used measure in the Douglas-fir region. 

RDsum = 0.01039 × SDIsum is a close approximation over the range of the 

present data.
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RESEARCH PAPER PNW-RP-607

In this even-aged plantation, SDIsum values were, as expected, close to those 

for Reineke SDI, being about 3 percent lower in 2013. Likewise, there is little difference at present between RDsum and RDqmd. However, the widest spacings are 

beginning to develop a naturally established understory, and they will soon contain 

many trees greater than the 4.0 cm lower measurement limit used here. Inclusion of 

these will in the near future materially affect the diameter distributions and QMD. 

Therefore, it seemed preferable to use one of the summation forms above rather 

than one that is dependent on QMD, even though there is little difference at this 

point in time.

Maximum Relative Density 

It is generally considered that an upper limit of SDI (or other relative density measure) exists and is characteristic of a species. Reineke (1933) estimated maximum 

SDI as SDIqmd = 595 (1470 in metric units) for Douglas-fir.

In the 2013 data, the highest values observed on any individual plot (plot 37) 

were SDIsum = 1367 and SDIqmd = 1421. Among spacing means, maxima were 

SDIsum = 1191 and SDIqmd = 1245 in the 1 × 1 m spacing. Similarly, RD maxima 

among spacing means were RDsum = 12.89 and RDqmd = 13.58 in the 1 × 1 m 

spacing. These values (table 4) are broadly consistent with maximum density 

estimates from other sources.

Because the maximum is not a precisely determined value, we have preferred 

to use the calculated values of SDIsum in our comparisons, rather than expressing 

stocking as a percentage of a maximum which itself is only an approximation.

Volume

Figure 11 shows that in 2013, volume in trees ≥4.0 cm peaked at 2 m spacing, and 

that in trees ≥20 cm at about 4 m spacing. Harrington et al. (2009) concluded that 

the 10-ft spacing (~3 m) in common use is near optimal. This may change if stands 

are allowed to develop further without thinning. The CV20 curve is expected to 

approach the CV4 curve over time, and conclusions about the “optimal” spacing 

may well change with advancing age. If stands were allowed to grow for a substantial additional period without thinning, it seems likely that 4 m (~13 ft) or greater 

spacing would become superior.
17

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Table 6— Fit statistics for some alternative Live Crown Ratio equations, 2013 data

Equation form RSMEa Adjusted RSQ

LCR100 = a + bSDIsm + c/SDIsum + d(SDI)0.5 0.0413 0.9116

LCR100 = a + bBA + c/BA + d(BA)0.5 0.0467 0.8866

LCRqmd = a + bSDIsum + c/SDIsum + d(SDI)0.5 0.0397 0.9467

LCRqmd = a + bBA + c/BA + d(BA)0.5 0.0544 0.9010

a Root mean square error.

Live Crown Ratio (LCR) and Height to Live Crown (HLC)

For each measurement on each plot, live crown ratio was regressed on D. The 

resulting equations were then entered with D100 and QMD to produce estimates of 

corresponding live crown ratios by spacing. Estimates for the 2013 measurement 

(total age 35) are compared in figure 14. Corresponding estimates of HLC100 

[= H100 – ( LCR × H100)] are shown in figure 2. As would be expected, values are 

strongly related to initial spacing. Differences between LCR100 and LCRqmd are 

greatest in the 1 × 1 m spacing, and decrease as spacing increases.

Figure 15 shows LCR100 trends over time and spacing. 

There are strong relationships at the 2013 measurement between LCR100 and 

SDIsum (fig. 16), and between LCRqmd and SDIsum. There are similar relationships of LCR100 and LCRqmd with basal area. These can be described by regressions of the form:

LCR = a + bX + c/X + dX0.5

Fit statistics for the above regression (table 6) suggest that SDIsum may be a 

slightly better predictor than basal area, although differences are small.

Figure 17 shows trends of LCR100 in relation to time and SDIsum. The pattern for LCRqmd is similar. Values for the 1 m spacing are very different from the 

others, and those for the 2 m spacing somewhat so. These differences probably 

reflect difficulty in identifying and measuring height to sparse live crowns in very 

dense stands, in addition to density effects. If we consider only spacings of 3 m and 

greater, the time trends suggest that SDIsum (or equivalent relative density measure) should be a good predictor of live crown ratio.

Figure 18 is a similar plot of LCR100 in relation to time and basal area. 

Compared to figure 17, this appears to show a slight superiority of basal area over 

SDIsum as a predictor.
18

RESEARCH PAPER PNW-RP-607

Figure 18—Time trends of LCR100 in relation to spacing and mean values of basal area at each 

measurement. Ages 17, 21, 27 and 35 years from seed.

Figure 17—Time trends of LCR100 in relation to spacing and mean values of SDIsum at each 

measurement. Ages 17, 21, 27 and 35 years from seed.

0 200 400 600 800 1000 1200 1400

SDIsum

1 m

2 m

3 m

4 m

5 m

6 m

0

0.2

0.4

0.6

0.8

1.0

1.2

L

C

R100

0

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50

L

C

R100

Basal area (square meters per hectare)

1 m

2 m

3 m

4 m

5 m

6 m
19

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 19—Spacing means of estimated ∑CA/ are, at 33 years from planting, 35 from seed, ± one 

standard error.

Crown Width and Crown Projectional Area

Table 4 and figure 19 show that variability in plot estimates of ∑CA/ha is greater in 

the 1 m and 2 m spacings than in other spacing. This reflects the greater mortality 

at these spacings as well as probable greater relative errors in measurement. They 

also show that the estimated spacing sums of crown projectional area are more than 

twice the corresponding ground areas for the 1 m and 2 m spacings. Such values are 

not plausible. We think these very high estimates are the combined result of several 

factors, including: (1) measurement of CW to the tip of the longest live branch, 

which probably overestimates effective CW and CA, (2) intermingling of branches 

at very close spacing, which makes it difficult to identify and measure to the 

margins of crowns, and (3) the fact that a given absolute error in determining crown 

width results in a relative error in estimated crown projectional area that is much 

larger for small narrow-crowned trees than for larger wide-crowned trees. 

We note that Gill et al. (2000) used a somewhat similar summation of estimated 

crown areas procedure to obtain apparently reasonable crown cover estimates. 

However, their procedure differed in that they measured crown radii to the esti0

0.50

1.00

1.50

2.00

2.50

1 2 3 4 5 6

sum

C

A/ha

Spacing (m)
20

RESEARCH PAPER PNW-RP-607

mated intersection of adjacent overlapping crowns rather than to the tip of the 

longest branch. This difference would give substantially lower estimates of ∑CA.

We also note that Reukema and Smith (1987) found average crown widths 

considerably greater than the average distance between trees in plots planted to 0.9 

m and 1.8 m spacing. Presumably this also can be attributed to a combination of 

actual crown overlap and error arising from the difficulty of defining and measuring crown margins in very dense stands of small-crowned trees.

Others have encountered similarly unreasonable estimates. Crookston and 

Stage (1999) provided a method for adjusting for crown overlap. Their equation is: 

% crown cover = 100[1 – exp(-0.01 C’)], 

where C’ is ∑(crown area calculated from crown width)/ground area. They assumed 

random spatial distribution, whereas our data had a uniform spatial distribution 

and would therefore be expected to have less crown overlap. We reasoned that 

there should be little overlap at the widest spacing (6 m), and therefore modified the 

coefficient in their equation from 0.01 to 0.018, which causes the estimate to coincide with the mean of the 6 m spacing. Figure 20 compares the result with values 

predicted by the unmodified equation; the modified estimates appear much more 

reasonable.

If the unmodified values for the 1 m and 2 m spacings are omitted, the other 

values fall in a well-defined line with relatively little scatter (fig. 21). These indicate 

that with near-uniform plantation spacing, full crown closure, 1.0, occurs at about 

SDIsum = 400, which is about one-third of maximum SDIsum. Similarly, closure is 

at about RDsum = 4, which again is about one-third of the possible maximum. The 

scatter of points is considerably less than in a plot of ∑CA/(plot area) over initial 

spacing.

Figure 22 is a plot of the time trends of means of ∑CA/(plot area), by spacing, 

over spacing means of SDIsum. Figure 23 is a similar plot over spacing means of 

basal area. (The 1 m spacing is omitted because of the very erratic trends associated 

with mortality and the difficulties with crown measurements noted above.) The 

figures suggest that SDIsum is a better predictor of ∑CA/plot area than is basal area 

in this unthinned plantation, contrary to the result of the comparison of LCR100 in 

figures 17 and 18. If we include all spacings, the simple correlation between spacing 

means of ∑CA/plot area and corresponding means of SDIsum is r = 0.95, vs. 0.85 

for that with mean basal areas. If the 1 m spacing is excluded, these values become 

r = 0.98 vs. 0.94.
21

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Figure 20—Crown cover estimates from (1) equation 2 in Crookston and Stage (1999) and (2) from 

equation modified to coincide with sum of calculated crown areas per unit area in the 6 m spacing, 

plotted over spacing means of SDIsum.

Figure 21—Individual plot estimates of ∑CA/ area, in relation to plot SDIsum, 33 years from planting, 35 from seed; 1 m and 2 m spacings omitted.

0

20

40

60

80

100

120

0 200 400 600 800 1000 1200 1400

Cro

wn co

ver (percent)

SDIsum

Crookston and Stage (1999)

Modified equation

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 100 200 300 400 500 600 700 800 900

sum

C

A/ha

SDIsum
22

RESEARCH PAPER PNW-RP-607

Figure 22—Time trends of spacing means of estimated ∑CA/ area in relation to corresponding 

means of SDIsum; 17, 21, 25, and 35 years from seed.

Figure 23—Time trends of spacing means of estimated ∑CA/ area in relation to corresponding 

mean basal areas; 19, 23, 27, and 35 years from seed.

2.5

2.0

1.5

1.0

0.5

0

sum

C

A/ha

2 m

3 m

4 m

5 m

6 m

0 200 400 600 800 1,000 1,200

SDIsum

2.5

2.0

1.5

1.0

0.5

0

0 5 10 15 20 25 30 35 40 45 50

sum

C

A/ha

Basal area (m2/ha)

2 m

3 m

4 m

5 m

6 m
23

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Conclusion

Strong points of this study are excellent plot replication within the study area and 

reliable measurements, except that the crown width samples are undesirably small. 

Otherwise, the main limitation of the study is simply that it is confined to a single 

physical location, which limits generalization of results. The data should be very 

useful in future growth modelling efforts when combined with data from other past 

and ongoing spacing studies.

This report was motivated in part by the need to update the information given 

by Harrington et al. (2009), with the additional 8 years of observation. There are no 

surprises in the usual stand statistics, which follow trends that are generally consistent with observations from other spacing trials.

Motivation for analysis of the crown development data arose in part from 

interest in relative density measures as possible estimators of crown cover and shading, particularly in relation to forest practice rules governing treatment of stands 

in riparian zones. In Washington, Oregon, and Idaho, the existing rules specify 

acceptable stocking in terms of basal area. 

It is here shown that for a range of initial planting spacing on the experimental 

area, live crown ratio is also related to SDIsum (and consequently, to the other 

closely related relative density measures), although differences from the corresponding and slightly closer relationship with basal area are small. Estimates of 

∑CA are also related to relative density, and in these data have a somewhat closer 

relationship with SDsum than with basal area.

Relative density measures can be useful estimators of average crown characteristics and shading (e.g. Teply et al. 2014), and are more readily obtained than 

direct measurements of crown cover. They have a potential advantage over basal 

area in that—unlike basal area—they are not inherently related to average tree size 

or to site and age (Jack and Long 1996; Long 1985; Reineke 1933). Hence, their use 

would not require an estimate of site index or site class. 

Differences in predictive ability of basal area vs. relative density in this study 

are small, because both are age-related and no site difference is present. Results 

in this study are suggestive, but the hypothesis that relative density measures can 

provide better estimates of crown cover than basal area cannot be confirmed or 

refuted on the basis of this study, because the data represent only one site and stand 

structure (pure plantation), and the same very limited range in age. It is highly 

desirable that the relationships discussed here be examined in other data including a 

range in site quality and age.

Crown width samples in any future work should be considerably larger than 

in the 2013 measurement in this study. Crown cover estimates from large-scale 
24

RESEARCH PAPER PNW-RP-607

aerial photos or LiDAR imaging may be a better alternative, if these can be paired 

with concurrent ground-based basal area and relative density estimates at the same 

locations.

English Equivalents

1 centimeter = 0.3937 inch

1 meter = 3.2808 feet

1 square meter = 10.76 square feet

1 hectare = 2.47 acres

1 square meter per hectare = 4.36 square feet/acre

1 cubic meter per hectare = 14.3 cubic feet/acre

SDIsum in metric units = 2.47 × SDIsum in English units

RDqmd in metric units = RDqmd in English units/6.9487

References

Bruce, D.; DeMars, D.J. 1974. Volume equations for second-growth Douglasfir. Res. Note PNW-239. Portland, OR: U.S. Department of Agriculture, Forest 

Service, Pacific Northwest Research Station. 5 p.

Crookston, N.L.; Stage, A.R. 1999. Percent canopy cover and stand structure 

statistics from the Forest Vegetation Simulator. Gen. Tech. Rep. RMRS-GTR-24. 

U. S. Department of Agriculture, Forest Service, Rocky Mountain Research 

Station. 11 p.

Curtis, R.O. 1971. A tree area power function and related stand density measures 

for Douglas-fir. Forest Science. 17(2): 146–159.

Curtis, R.O. 1982. A simple index of stand density for Douglas-fir. Forest Science. 

28: 92–94.

Curtis, R.O. 2010. Effect of diameter limits and stand structure on relative density 

indices: a case study. Western Journal of Applied Forestry. 25(4): 169–175.

Franklin, J.F.; Dyrness, C.T. 1973. Natural vegetation of Oregon and Washington. 

Gen. Tech. Rep. PNW-8. Portland, OR: U. S. Department of Agriculture, Forest 

Service, Pacific Northwest Forest and Range Experiment Station. 417 p.

Gill, S.J.; Biging. G.S.; Murphy, E.C. 2000. Modeling conifer tree crown radius 

and estimating canopy cover. Forest Ecology and Management. 126: 405–416.
25

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Harrington, T. B.; Harrington, C.A.; DeBell, D.S. 2009. Effects of planting 

spacing and site quality on 25-year growth and mortality relationships of 

Douglas-fir (Pseudotsuga menziesii var. menziesii. Forest Ecology and 

Management. 258(10): 18–25.

King, J.E. 1966. Site index curves for Douglas-fir in the Pacific Northwest. For. 

Pap. 8. Centralia, WA: Weyerhauser Forestry Research Center. 49 p.

Jack, S.B.; Long, J.N. 1996. Linkages between silviculture and ecology: an 

analysis of density management diagrams. Forest Ecology and Management. 

86: 205–220.

Long, J.N. 1985. A practical approach to density management. Forestry Chronicle. 

61(1): 23–27.

Long, J.N.; Daniel, T.W. 1990. Assessment of growing stock in uneven-aged 

stands. Western Journal of Applied Forestry. 5(3): 93–96.

Miller, R.E.; Reukema, D.L.; Anderson, H.W. 2004. Tree growth and soil 

relations at the 1925 Wind River spacing test in coast Douglas-fir. Res. Pap. 

PNW-RP-558. Portland, OR: U.S. Department of Agriculture, Forest Service, 

Pacific Northwest Research Station. 41 p.

Reineke, L.H. 1933. Perfecting a stand-density index for even-aged forests. Journal 

of Agricultural Research. 46: 627–638.

Reukema, D.L. 1978. Fifty-year development of Douglas-fir stands planted at 

various spacings. Res. Pap. PNW-253. Portland, OR: U. S. Department of 

Agriculture, Forest Service, Pacific Northwest Forest and Range Experiment 

Station, Portland. 21 p.

Reukema, D.L.; Smith, J.H.G. 1987. Development over 25 years of Douglasfir, western hemlock, and western redcedar planted at various spacings on a 

very good site in British Columbia. Res. Pap. PNW-RP-381. Portland, OR: 

U.S. Department of Agriculture, Forest Service, Pacific Northwest Research 

Station. 46 p.

Teply, M.; McGreer, D.; Ceder, K. 2014. Using simulation models to develop 

riparian buffer strip prescriptions. Journal of Forestry 112(3): 302–311.

Wilson, J.S.: Oliver, C.D. 2000. Stability and density management in Douglas-fir 

plantations. Canadian Journal of Forest Research. 30: 910–920.
26

RESEARCH PAPER PNW-RP-607

Acknowledgments

Helpful reviews were provided by David Marshall, Douglas Maguire, Dean DeBell, 

and Richard Miller. Dave Thornton maintained and edited the data.

Glossary of Symbols and Acronyms

CV4 = cubic volume in trees of D ≥ 4 cm.

CV20 = cubic volume in trees of D ≥ 20 cm.

D = dbh = diameter at breast height.

D100 = mean diameter of the 100 largest (by diameter) trees per hectare.

di = dbh of an individual tree “i”.

H100 = mean height of the 100 largest (by diameter) trees per hectare.

HLC = height to base of live crown.

Hqmd = height corresponding to quadratic mean diameter.

LCR = live crown ratio.

LCR100 = mean LCR of the 100 largest (by diameter) trees per hectare.

LCRqmd = LCR corresponding to the quadratic mean diameter.

N = number of trees per unit area.

QMD = quadratic mean diameter.

RDqmd = (Basal area)/sqrt(QMD).

RDsum (in metric units) = (0.00007854) × ∑di

3/2 / area.

SE = standard error of the mean.

SDIqmd = Reineke stand density index, calculated from QMD and number of trees.

SDIsum = ∑(di/25.4)1.6 / area.
27

Relation of Initial Spacing and Relative Stand Density Indices to Stand Characteristics in a Douglas-fir Plantation Spacing Trial 

Appendix

Table 7—Spacing means at 2013 remeasurement, 33 years from planting, 

in English units

Spacing H100 D100 QMDa Numbera Basal areaa

 Volume 

CV4a

Volume 

CV20b

Meters 

(Feet)

Feet Inches Inches N/acre Square 

feet/acre

Cubic 

feet/acre

Cubic 

feet/acre

1 (3.3) 68.6 9.1 4.4 1,855 199 4,306 728

2 (6.6 73.6 10.6 6.7 775 190 4,982 2491

3 (9.8) 72.8 11.8 7.4 400 144 3,791 2,889

4 (13.1) 74.5 13.2 9.8 231 122 3,270 3,030

5 (16.4) 72.9 13.4 10.7 159 100 2,615 2,508

6 (19.7) 70.8 13.4 11.2 113 80 2,001 1,940

a Trees ≥ 1.6 in.

b Trees ≥ 7.9 in.

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