Height–diameter models for tropical forests on Hainan Island in southern China

Forest Ecology and Management 110 (1998) 315±327

Height±diameter models for tropical forests on Hainan Island in southern China Zixing Fang, R.L. Bailey* Warnell School of Forest Resources, The University of Georgia, Athens, GA 30605, USA Received 30 September 1997; accepted 24 March 1998

Abstract Measurements on 8352 felled trees from 31 plots in tropical forests on Hainan Island provided information to investigate the relationship between total tree height and diameter at 1.3 m (DBH). Out of 33 model forms either published or derivable from published forms, a modi®cation of the exponential model published by Meyer (Meyer, H.A. A mathematical expression for height curves. J. For. 38 (1940) 415±420) was selected based on low bias and relatively good precision of its predictions on individual plots. Variations from plot-to-plot were signi®cant. Therefore, a parameter prediction method based on predictor variables derived from all DBH values on a plot was developed in order to provide a method to localize the model. In a test with independent data from an additional 36 plots, the localized height-prediction model produced predictions with a pooled mean-squared error of 3.8 m2, a pooled R2 of 0.72, and an average bias of ÿ0.2 m. Even though tremendous variability exists in tropical forests and modeling height±DBH relationships never provides high levels of precision, this model will provide a useful and economical alternative to attempting height measurements during inventories of these forests. # 1998 Elsevier Science B.V. Keywords: Inventory; Stand structure; Nonlinear models

1. Introduction Considering that diameter at breast height (DBH) is relatively cheaper and can be more accurately obtained than total tree height and that a close relationship usually exists between DBH and height of individual tree stems (Mayer, 1936), foresters often turn to height-DBH regressions to predict total tree height based on observed DBH. When using one-entry volume tables or a standard volume table to estimate *Corresponding author. Tel.: 00 1 706 5421187; fax: 00 1 706 5428356; e-mail: [email protected] 0378-1127/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved. PII S0378-1127(98)00297-7

tree or stand volume, accurate height±DBH functions are usually required (Curtis, 1967; Tang, 1994). According to Tang (1994) the error in predictions with one-entry volume tables presently used in China mainly comes from the estimation error of the average heights in diameter classes. Thus, any advancement in the ability to precisely predict tree height from DBH could achieve signi®cant improvements in the precision of timber inventories, especially for a region where little work has been done on this topic. Hainan Island is such a place. Hainan Island is located at 188090 ±208100 N, 1088030 ±1118030 in southern China. The forests on

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Hainan Island belong to the family of tropical forests on its northern margin, which are quite different both in stand structure and in species composition from equatorial tropical forests (Hu, 1992). Very little forest research work has ever been done for this region, including the basic topic of height±diameter relationships. In tropical regions, the measurement of total tree height is quite a challenge due to the relatively large crown sizes and high stand density (Hu, 1992; Vanclay, 1994). This is perhaps one main reason that no published height±diameter functions exist for the tropical forest on Hainan. The primary objectives of our work are to analyze potential height±diameter functions as found in the forestry literature and present the most appropriate height±diameter equations and their properties for tropical forest on Hainan Island. 2. Data description Two sets of data were used in this study. The ®rst set consists of 8352 felled trees from 31 plots established in undisturbed tropical forests on Hainan Island (Table 1). Foresters collected these data during the 1950±60 period. Due to some historical reasons, forest researchers have rarely used them. Recognizing that much valuable information might be gleaned from them, we recently initiated an effort to record the data on computer media and develop mensurational models with them. There are a number of interesting and useful relationships to be explored, but in this report we will focus on the relationship between DBH and the total tree height. Since the data were collected after cutting down the trees, we assume that the total tree height and some other characteristics were more accurately measured for these trees than equivalent data collected from standing trees. Hence, we use this set of data to select, analyze and develop the models. The other set of data was collected in the Bawanglin forest region on Hainan Island in 1993 and consists of 36 plots randomly taken from that region (Table 2). These data were the basis of an independent test of the models selected and ®tted with the ®rst data set described above. Tropical forests are extremely rich in species but there is usually no predominant species. Our ®rst data set represented 217 distinct species (Table 3). The second data set was even richer and contained a total

of 232 species (Table 3). In neither data set did any one species represent more than 10% of the trees by either numbers or basal area. 3. Methods and modeling approach There are a great number of references about height±diameter relationships for different species and different forest regions. Curtis (1967) compared 13 height±diameter equations using linear regression techniques and data on second growth Douglas-®r. Huang and Titus (1992) selected the most appropriate height±diameter functions for major tree species in their study from 20 nonlinear equations by weighted nonlinear techniques. Mayer (1936) made suggestions concerning appropriate characteristics for height±diameter curves that have had a lasting effect on forest modelers. In addition to the requirement that the model should be moderately ¯exible, he proposed that it should possess the following characteristics (see Curtis, 1967): 1. The slope should be positive everywhere, and should approach zero as the DBH becomes large. 2. The curve should pass through the origin. 3. The model should be easily fitted by the linear regression methods. When point (1) is true, no reduction in height with increasing DBH occurs for an individual tree. To invoke this constraint usually requires us to omit data for any trees with broken tops when we develop height±diameter equations. Since the number of broken tops is usually low, this causes little loss of data. Point (1) also suggests that some form of growth equation with age, replaced by DBH, may be appropriate to model the height±diameter relationship. Point (2) is still an important constraint for many forest modelers. To invoke it, they either form a variable by subtracting 1.3 m from total tree height or add a constant 1.3 to the right side of height± diameter equations to meet a theoretical assumption that the tree DBH should be zero when the total tree height is 1.3 m. The assumption is theoretically reasonable, since the DBH is de®ned as the tree diameter at breast height (1.3 m above ground). But, in practice, seldom have people really been interested in trees so small as when DBH equals zero. In sample data for

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

317

Table 1 Basic statistics for Data Set 1 a. Total tree height and DBH statistics from 31 plots in tropical forest stands on Hainan Island in Southern China Plot

1 2 3 4 5 6 7 8 101 102 103 104 105 106 107 108 201 202 203 301 302 303 304 305 306 307 308 310 311 888 999

Area (ha)

Stems

0.240 0.260 0.260 0.260 0.260 0.300 0.280 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.175 0.250 0.200 0.190 0.206 0.240 0.210 0.240 0.170 0.565 2.800

185 240 247 128 209 299 229 208 166 158 140 160 176 172 178 161 164 172 150 177 149 309 221 215 197 280 167 274 178 426 2217

Sp a

50 59 59 44 38 43 55 44 55 49 49 29 37 36 47 49 51 47 37 52 46 43 60 44 47 45 35 36 30 70 101

DBH (cm)

Total tree height (m)

mean

std.

min.

max.

21.0422 17.2929 20.8822 19.6602 19.8512 18.3572 21.0393 17.0091 19.5181 22.2222 21.7836 18.8631 18.6756 18.6331 20.9152 25.3888 20.8293 20.6110 22.2473 20.2362 19.8718 17.3667 19.6416 20.2377 20.9929 17.2057 19.6144 16.0270 17.9697 19.1556 20.3548

11.7240 9.6791 11.5518 10.0768 11.3861 10.3179 12.3232 9.0953 11.9727 17.6830 12.1903 12.5900 10.5834 12.3004 12.7489 19.9265 11.1269 13.9190 15.7950 15.4284 11.7265 10.7200 14.3972 15.1784 12.2807 11.7827 10.3054 8.6459 9.7877 12.1974 13.4341

8.0 8.0 8.0 8.0 8.0 8.0 8.0 7.5 8.3 8.3 8.0 8.0 8.0 8.0 8.0 8.0 8.1 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.5 8.0 8.0 8.0 8.2 8.0 8.0

65.0 86.2 92.5 61.7 59.3 58.5 72.0 66.2 77.9 130.0 73.5 68.0 62.4 75.3 83.2 113.0 69.0 71.2 100.5 150.0 72.5 67.7 112.0 136.0 78.2 77.5 60.1 61.0 55.7 78.6 174

QMD

b

24.0724 19.8076 23.8531 22.0742 22.8712 21.0497 24.3690 19.2779 22.8787 28.3643 24.9412 22.6569 21.4511 22.3073 24.4758 32.2365 23.5990 24.8481 27.2537 25.4203 23.0538 20.3997 24.3338 25.2760 24.3054 20.8416 22.1425 18.2029 20.4492 22.7017 24.3867

c

2

BA (m ) 8.420 7.395 11.038 4.899 8.586 10.405 10.681 6.071 6.824 9.984 6.840 6.451 6.361 6.722 8.375 13.140 7.173 8.341 8.750 8.983 6.220 10.099 10.278 10.788 9.140 9.552 6.431 7.131 5.846 17.243 103.553

mean

std.

min.

max.

14.9741 13.5979 15.8364 15.0539 13.7220 12.5445 15.7782 13.0615 14.9970 15.5367 16.9114 12.7519 14.0540 14.1000 15.5028 17.2925 16.1433 15.7000 16.7513 15.3350 16.4926 15.8660 14.8330 12.7777 14.5553 15.2443 13.7563 12.8467 12.7826 15.4155 16.2756

4.42877 3.45913 4.10019 3.85972 3.76908 3.34017 4.35762 3.19877 4.59900 4.94325 4.49194 3.80357 3.75455 4.33057 4.77599 6.44649 4.58493 4.83908 5.13216 4.53271 4.59330 4.53256 4.70524 4.63781 4.21089 5.13831 3.52348 3.04746 3.35907 4.64271 5.23476

6.8 6.0 5.7 7.4 6.5 5.5 7.8 5.4 7.4 5.2 7.3 6.0 6.0 7.0 7.4 7.0 7.2 6.0 7.4 8.0 9.0 5.9 6.3 5.4 5.2 7.0 6.4 6.4 5.0 5.8 5.0

27.9 25.3 28.0 25.6 23.6 23.5 28.0 25.4 30.7 33.0 27.0 27.8 22.8 26.8 35.7 33.5 30.0 29.3 30.2 30.0 29.9 28.8 31.0 29.0 27.3 30.4 24.0 25.0 23.4 31.0 32.4

a

Total basal area of the plot (m2). Total number of distinct species. c Quadratic mean diameter (cm). b

forest growth and yield modeling, we usually measure the trees whose DBH values are larger than a certain lower limit (such as 5 or 8 cm). Thus, arbitrarily assuming (0,1.3) for a (DBH, height) data pair may not represent any of our data. Statistically, it makes little, if any, sense to predict the height by some value of DBH that is beyond the range of the data used to develop the model. In our case, the minimum DBH class measured was 8 cm in Data Set 1 and 5 cm in Data Set 2. So, we have elected not to use (0,1.3) as a constraint on (DBH, height) in our models. Since regression software packages (both linear and nonlinear) are now readily available and relatively

easy to obtain, point (3) is obviously less important than in Mayer's day. We relied heavily on the SAS software for our analysis (SAS Institute Inc., 1996). The equations selected for comparison here were based on an examination of the height±diameter relationship as revealed by plotting tree height against DBH and the hypothesis that height±diameter relationships can be described by height growth curves with age replaced by DBH. Some equations referenced here are for discussion purposes. Table 4 provides a complete list of the functions selected for comparison.

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Table 2 Basic statistics for Data Set 2 a. Total tree height and DBH statistics from 36 plots randomly located in tropical forest stands in Bawanglin forest on Hainan Island in southern China Plot

1 3 4 5 7 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Stems

45 130 128 63 56 41 54 73 81 88 58 78 111 113 77 60 99 87 67 86 93 78 97 76 104 136 47 84 121 105 95 38 157 146 85 82

BA b

0.84772 1.99142 1.20397 2.56864 1.80364 1.74774 2.45499 2.41202 1.62669 0.76885 1.33185 3.01286 1.93058 4.54436 3.36652 1.79910 1.57788 3.26761 1.26372 2.34570 2.40145 2.70546 4.94147 1.96798 1.86127 2.46164 2.22827 3.08289 2.14872 1.69430 2.59997 2.02658 2.11077 1.86961 2.53201 2.56977

QMD c

15.4873 13.9658 10.9435 22.7844 20.2505 23.2971 24.0593 20.5109 15.9906 10.5471 17.0989 22.1767 14.8812 22.6283 23.5939 19.5392 14.2454 21.8681 15.4968 18.6356 18.1322 21.0149 25.4681 18.1576 15.0954 15.1809 24.5691 21.6169 15.0367 14.3336 18.6671 26.0583 13.0835 12.7689 19.4750 19.9754

DBH (cm)

Total tree height (m)

mean

std

min

max

mean

std

min

max

13.4422 11.0531 9.9453 15.5206 16.7786 19.3585 20.1259 16.1603 13.8432 9.6886 15.1103 16.7372 12.0838 16.4903 17.2455 16.0567 12.2384 18.5425 12.7179 15.0012 15.5204 16.7474 20.6165 14.1868 12.3692 12.6412 18.6638 18.9690 12.7554 11.3010 15.3758 21.3158 10.5433 11.2260 15.8153 14.7512

7.7787 8.5695 4.5843 16.8144 11.4412 13.1226 13.3070 12.7184 8.0540 4.1919 8.0730 14.6432 8.7245 15.5646 16.2074 11.2279 7.3277 11.6598 8.9216 11.1215 9.4259 12.7767 15.0305 11.4081 8.6947 8.4370 16.1510 10.4289 7.9957 8.8593 10.6414 15.1902 7.7719 6.1055 11.4320 13.5520

5.0 5.0 5.0 5.1 5.1 6.6 5.6 5.1 5.1 5.0 5.5 5.1 5.0 5.1 5.3 5.3 5.0 5.2 5.1 5.4 5.2 5.3 5.9 5.6 5.0 5.1 5.5 5.4 5.1 5.0 5.1 5.6 5.0 5.0 5.7 5.0

37.4 62.1 28.0 92.0 56.3 57.5 72.0 82.4 61.1 23.7 40.7 104.3 70.7 97.7 84.2 62.0 58.6 82.8 44.0 59.2 47.7 81.1 68.0 70.0 51.8 50.9 77.0 50.2 44.3 46.2 46.5 63.7 59.5 45.1 82.8 97.1

8.5244 8.8715 8.0141 9.2254 11.9643 11.7268 12.0704 11.1233 10.8765 6.8080 10.2086 10.5756 8.5225 9.9956 10.1286 11.2083 9.3232 11.5138 9.1269 10.2000 10.0118 11.7077 13.3454 9.7671 9.5808 9.7404 10.9383 11.7095 9.9669 8.9895 10.2958 11.3289 9.5115 9.6726 10.2247 9.8171

2.72281 2.86852 1.73794 3.70470 5.13974 4.48643 3.30098 5.34136 3.14298 1.49284 2.86315 3.46812 4.08956 4.59546 3.93927 4.25888 3.39140 4.26250 3.41603 4.66980 3.72958 3.52655 5.40272 3.07686 2.64300 2.72044 5.13354 3.16074 3.10911 3.60736 3.65673 4.39944 3.62963 3.67817 3.24224 4.36492

4.7 5.0 4.9 4.0 3.0 5.5 4.8 4.0 5.0 4.8 5.0 5.0 3.0 5.0 4.5 4.0 4.0 3.0 5.0 5.0 4.0 4.7 5.5 4.6 5.6 4.0 4.8 4.9 5.6 3.8 5.2 4.0 4.5 4.0 4.8 4.5

14.8 19.0 12.8 21.0 26.5 19.0 19.0 26.0 22.0 12.5 16.6 22.5 25.0 24.0 22.0 20.5 20.0 24.0 18.0 25.0 18.0 25.8 26.5 21.0 17.5 19.0 26.8 18.8 18.0 20.8 18.8 24.0 24.0 24.0 22.5 22.0

a

The area of each plot is 0.067 ha. Total basal area of the plot (m2). c Quadratic mean DBH (cm). b

It usually is desirable to model the height±diameter relationship by distinct tree species. However, for tropical forests, it is hard to ®nd a dominant species for most stands (Table 3). Besides, in the present forest practices on Hainan Island, information on species is usually not available in an ordinary forest inventory (such as in CFI). Therefore, we chose not to model height±diameter relationships for distinct spe-

cies in this study. This does not mean that species is unimportant in height±diameter modeling. In fact, more precise regression models may possibly be obtained with separate height±diameter models for each species. They would be of limited use, however, when foresters recording data in the ®eld are unable to determine the species of each tree with ease and reliability.

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

319

Table 3 The 10 most frequent species represented in the two sets of data from tropical forest stands on Hainan Island in southern China a Data Set 1

Data Set 2

species

frequency

Cryptocarya chinensis Castanopsis tonkinensis Gironniera subaequalis Alseodaphne hainanensis Syzygium sp. Symplocos sp. Machilus sp. Adinandra hainanensis Xanthophyllum hainanensis Linociera sp. Total

6.9% 6.5% 4.8% 4.4% 4.3% 4.1% 3.8% 2.4% 2.3% 2.3% 41.8%

a b

b

species

frequency b

Castanopsis tonkinensis Symplocos sp. Syzygium sp. Camellia sinensis Xanthophyllum hainanensis Cryptocarya chinensis Ilex sp. Diospyros eriantha Helicia formasana Linociera sp. Total

9.4% 5.9% 5.3% 4.0% 3.6% 3.6% 2.9% 2.6% 2.4% 2.4% 42.1%

There were 217 distinct species for the 8352 felled trees of Data Set 1 and 232 distinct species for the 3139 trees of Data Set 2. Frequency as percent of the total number of stems.

Table 4 HEIGHT±DBH models selected for comparison with data from tropical forest stands on Hainan Island in southern China Model number and form a

References

Linear equations: [1] H ÿ1 ˆ a ‡ bDÿ1 [2] II ˆ a ‡ b log D [3] log H ˆ a ‡ b log D [4] log H ˆ a ‡ bDÿ1 [5] H ˆ a ‡ bD ‡ cD2 [6] H ˆ a ‡ bDÿ1 ‡ cD2

Vanclay (1995) Curtis (1967), Alexandros and Burkhart (1992) Prodan (1965), Curtis (1967) Curtis (1967) Henricksen (1950), Curtis (1967) Curtis (1967)

Nonlinear equations: Two-parameter equations: 1. power equation: [7] H ˆ aDb 2. exponential equations: [8] H ˆ ea‡b=…D‡1† [9] H ˆ aeb=D [10] H ˆ a…1 ÿ eÿbD †

Wykoff et al. (1982), Huang and Titus (1992) Burkhart and Strub (1974), Burk and Burkhart (1984), Buford (1986) Meyer (1940), Farr et al. (1989), Moffat et al. (1991)

Hyperbolic equations: [11] H ˆ aD=…b ‡ D† [12] H ˆ D2 =…a ‡ bD†2 [13] IIˆa‡(aÿ1.3)*b/(D‡b)

Tang (1994), Bates and Watts (1980), Ratkowsky and Reedy (1986) Loetsch et al. (1973), Huang and Titus (1992) By redefining parameter b from [20]

Three parameter equations: 1. power equation: ÿc [14] H ˆ aDbD 2. exponential equations: c [15] H ˆ ea‡bD [16] H ˆ aeb=…D‡c† [17] H ˆ a ‡ b…1 ÿ eÿcD † [18] H ˆ a ‡ b…1 ÿ eÿc…DÿDmin † †

Stoffels and van Soest (1953), Stage (1975), Huang and Titus (1992)

Sibbesen (1981), Huang and Titus (1992) Curtis et al. (1981), Larsen and Hann (1987), Wang and Hann (1988) Ratkowsky (1990), Huang and Titus (1992) By adding one parameter from [10] By subtracting a constant term Dmin.

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Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

Table 4 (Continued ) Model number and form a 3. hyperbolic equation: [19] H ˆ D2 =…a ‡ bD ‡ cD2 † [20] Hˆa‡b/(D‡c) 4. monomolecular equation: [21] H ˆ a…1 ÿ ceÿbD † [22] H ˆ a…1 ÿ cÿb…Dÿe† † 5. Gompertz equation: [23] H ˆ aeÿb;exp…ÿcD† [24] H ˆ aeÿexp…ÿb…Dÿc†† 6. logistic equation: [25] H ˆ a=…1 ‡ beÿcD † [26] H ˆ a=…1 ‡ eÿb…Dÿc† † [27] H ˆ a=…1 ‡ bÿ1 Dÿc † 7. Weibull equation: [28] H ˆ a…1 ÿ expfÿbDc g† 8. Chapman±Richards equation: [29] H ˆ a…1 ÿ ebD †c Four parameter equations: 1. exponential equation (Sloboda): [30] H ˆ a expfÿbexp‰ÿcDd Šg 2. Chapman±Richards equation: [31] H ˆ a…1 ÿ beÿcD †d 3. Weibull equations: [32] H ˆ a…1 ÿ b expfÿcDd g† [33] H ˆ a…1 ÿ expfÿb…D ÿ c†d g† a

References Curtis (1967), Prodan (1968), Huang and Titus (1992) Tang (1994)

Winsor (1932), Huang and Titus (1992) Seber and Wild (1989) Pearl and Reed (1920), Huang and Titus (1992) Seber and Wild (1989) Ratkowsky and Reedy (1986) Bailey (1979) Richards (1959), Huang and Titus (1992)

Zeide (1992) Richards (1959), Huang and Titus (1992) Bailey (1979), Seber and Wild (1989) Seber and Wild (1989)

H total tree height (m); DˆDBH (cm); a, b, c, and d are parameters to be estimated; and e the base of the natural logarithm.

All of the ®rst set of data (8352 stems) was used to estimate the parameters of each model in Table 4. The SAS procedure REG was applied to all linear equations and to some others after appropriate data transformation. For example, for model [1]1 in Table 4 the DBH and total tree height were ®rst inverted to form two new variables (Y, heightÿ1; and X, DBHÿ1) before the use of the REG procedure to estimation the parameters. The SAS procedure NLIN was used to estimate parameters in nonlinear regression models. Since a closed form solution generally does not exist for nonlinear equations, the NLIN procedure uses an iterative process that depends on starting values for the parameters of the model being provided. The parameter values are re-estimated and improved until the error sum-of-squares changes less than some 1

Model reference numbers in square brackets (for example, [1]) refer to equations listed in Table 4. Numbers in parentheses, such as (1), refer to equations shown within the text.

speci®ed amount. There are three main iterative methods available in NLIN. They are the Gauss±Newton, Marquardt, and the steepest decent methods. The Marquardt method is a compromise between the Gauss±Newton and the steepest decent (Marquardt, 1963) methods, and it is most useful when the parameter estimates are highly correlated. Additionally, it is believed that the Marquardt method sometimes works when the default method (Gauss±Newton) does not. We elected to use the Marquardt method. Since scatter plots of the studentized residuals appeared to indicate homogenous variance for most equations tested with our data, we used the least squares rather than weighted least squares. Each model was ®tted with data from each plot in the ®rst data set. In order to use ®tted parameters from each plot±model combination to determine the most appropriate model for tropical forests on Hainan Island we determined average bias and precision in the following way.

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

321

Table 5 Parameter estimates for two-parameter height-diameter models fitted with data from tropical forest stands on Hainan Island in southern China Model a

Parameters

Performance criteria

a [1] [2] [3] [4] [7] [8] [9] [10] [11] [12] [13]

b

0.0325 b (0.00038) c ÿ7.052 (0.1502) 1.270 (0.010) 3.238 (0.0048) 3.967 (0.0374) 3.336 (0.0041) 27.28 (0.1091) 24.65 (0.138) 33.62 (0.2205) 1.229 (0.0088) 35.1 (0.254)

s2

MS

MSE

0.5824

0.1497

0.3792

2.1369

0.0000

0.1477

0.1477

2.0815

0.3737

0.1530

0.2086

2.1588

0.4281

0.1574

0.2443

2.2246

ÿ0.0153

0.1511

0.1523

2.1285

0.0198

0.1533

0.1549

2.1591

0.0233

0.1554

0.1576

2.1888

0.0514

0.1517

0.1625

2.1371

0.0126

0.1479

0.1491

2.0835

0.0223

0.1506

0.1529

2.1221

0.0077

0.1474

0.1484

2.0770

e

0.6051 (0.0052) 7.837 (0.0522) 0.4943 (0.0036) ÿ8.568 (0.0657) 0.4626 (0.0030) ÿ10.698 (0.0765) ÿ9.48 (0.0694) 0.048 (0.0005) 20.88 (0.2743) 0.1830 (0.0004) 24.9 (0.352)

a

See Table 3 for forms of the models. Estimated parameter. c Standard error of estimation. b

First, residuals are de®ned as: ^ ijk ; eijk ˆ Hijk ÿ H

(1)

^ ijk and eijk are the observed height, where Hijk and H predicted height, and residual error in height for the kth tree in the ith plot with model j. We de®ne the mean residual with model j for plot i as: eij ˆ

ni 1X eijk ni kˆ1

(2)

where ni is the number of trees on the ith plot. The mean residual, eij, was used as an indication of the model's prediction bias for the ith plot. A measure of precision was calculated for each plot and model combination as: Pni Pn i 2 2 kˆ1 eijk ÿ … kˆ1 eijk † 2 (3) sij ˆ …ni ÿ 1†ni A mean square error type of measure, combining both

the prediction bias and precision of a given model j ®tted to the data of a given plot i, was obtained as: MSij ˆ e2ij ‡ s2ij

(4)

The average values of eij, s2ij , and MSij over the 31 plots for each model along with the ordinary mean square error (MSEij) averaged over all 31 plots were used to compare model performance with our data. The parameter estimates for the two-, three-, and four-parameter models (see Table 4) and the average values of above measures of error are shown in Tables 5±7, respectively. 4. Results of model comparisons Several of the different models we chose produced very similar results with the criteria we considered. Upon re¯ection, this is not surprising. It can be explained from the fact that any nonlinear function

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Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

Table 6 Parameter estimates for three-parameter height±diameter models fitted with data from tropical forest stands on Hainan Island in southern China Model a

a [5] [6] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

Performance criteria d

Parameters b b

6.82 (0.076) c 29.36 (0.140) 43.2 (1.18) 4.31 (0.093) 35.04 (0.497) 4.35 (0.160) 9.67 (0.0598) ÿ1.34 (0.138) 37.74 (0.589) 29.1 (0.319) 29.1 (0.319) 27.13 (0.215) 27.13 (0.215) 26.06 (0.169) 26.06 (0.169) 43.6 (1.695) 33.5 (0.896) 31.4 (0.601)

0.498 (0.0052) ÿ327.2 (4.451) ÿ4.16 (0.077) ÿ3.98 (0.026) ÿ26.3 (0.958) 24.75 (0.232) 19.43 (0.291) 0.765 (0.015) ÿ1133 (65.4) 0.850 (0.0055) 0.0323 (0.0009) 1.473 (0.012) 0.0515 (0.001) 2.667 (0.0314) 0.0718 (0.0012) 0.0554 (0.0009) 0.079 (0.0012) 0.0213 (0.0014)

c ÿ0.0027 (0.00007) 1419.5 (31.45) 0.849 (0.0153) ÿ0.322 (0.021) 10.68 (0.570) 0.0323 (0.0009) 0.0323 (0.0009) 0.0269 (0.00035) 32.6 (1.56) 0.0323 (0.0009) ÿ5.02 (0.287) 0.0515 (0.001) 7.523 (0.141) 0.0718 (0.0012) 13.66 (0.197) 0.790 (0.0212) 0.708 (0.013) 0.627 (0.0123)

s2

MS

AMSE

0.0000

0.1490

0.1490

2.1050

0.0000

0.1491

0.1491

2.1058

0.0000

0.1466

0.1466

2.0699

0.0015

0.1467

0.1467

2.0713

ÿ0.0001

0.1465

0.1466

2.0698

0.0000

0.1468

0.1468

2.0721

0.0000

0.1468

0.1468

2.0721

0.0003

0.1465

0.1466

2.0697

ÿ0.0000

0.1466

0.1466

2.0697

ÿ0.0000

0.1468

0.1468

2.0721

0.0000

0.1468

0.1468

2.0721

ÿ0.0019

0.1475

0.1475

2.0812

ÿ0.0019

0.1475

0.1475

2.0812

ÿ0.0046

0.1484

0.1485

2.0939

ÿ0.0046

0.1484

0.1485

2.0939

0.0010

0.1466

0.1466

2.0702

0.0001

0.1466

0.1467

2.0702

0.0008

0.1466

0.1468

2.0703

e

a

See Table 3 for forms of the models. Estimated parameter. c Standard error of estimation. d Definitions are given in text of article; e is a measure of bias, s2 is a measure of overall precision, MSˆe2‡s2, and AMSE is the average mean square error over all 31 plots. b

may be expressed as a Taylor series expansion of a polynomial, truncated after terms become inconsequentially small. So, for models with the same numbers of parameters ®tted to the same set of data with the same nonlinear least-squares criteria for convergence being applied, upon convergence the above

statistics of ®t should be similar. Actually, some models compared have very close algebraic relationships to each other. For example, model [3] (see Table 4) is the logarithmic transformation of [7], and [20] a special case of [13]. Models [17], [18], [21] and [22] are all of the same form except for a

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

323

Table 7 Parameter estimations for four-parameter height±diameter functions Function

[30] [31] [32] [33]

Performance criteria a

Parameters a

b

c

d

e

n

MS

MSE

35.8 (2.85) 32.5 (1.15) 33.8 (1.90) 33.8 (1.73)

4.52 (1.49) 1.02 (0.012) 1.009 (0.046) 0.027 (0.003)

0.587 (0.230) 0.179 (0.028) 0.0825 (0.0180) 0.176 (0.867)

0.363 (0.086) 0.555 (0.048) 0.694 (0.072) 0.698 (0.053)

0.0018

0.1462

0.1465

2.0703

0.0000

0.1461

0.1461

2.0688

0.0005

0.1461

0.1461

2.0690

0.0006

0.1461

0.1461

2.0683

a There are 18, 12, 11 and 8 plots that failed to converge for functions [30], [31], [32] and [33], respectively, when the initial values were given by the respective estimated parameters from overall data.

difference in the initial point of the curve. Model [28] is a special case of [32], and [29] a special case of [31]. There are some important differences to be noted about the models. Models ®tted after a linearizing transformation (for example: [1], [3], [4]) have larger mean errors; such models tend to produce biased estimation due to the transformation error. Scatter plots revealed that the variance of model [1] is not homogeneous with our data, but the residuals of the nonlinear form of [1], namely Hˆ

D ; where D ˆ DBH and H ˆ height; D ‡ (5)

are nearly so. In the case of model [5], graphs showed very unreasonable height predictions for large diameters. Four-parameter models gain very little over three-parameter forms, either in prediction bias or precision. Moreover, the addition of a fourth parameter can cause a lot of trouble. Over-parameterization creates sensitivity to initial values and convergence is not so easily attained. Thus, we do not recommend any of the four-parameter models we tested for height±diameter equations. Since most of the models tested performed well as height±diameter equations for tropical forests on Hainan Island, the choice of a particular function may depend on the relative ease of achieving convergence (for nonlinear models), the equation's mathematical properties, and possible biological interpretations of parameters and predictions. For two-parameter mod-

els, the log-linear [2] and the nonlinear [13] models performed well considering ease of parameter estimation and relative precision in predictions. Of the three-parameter models tested, we recommend [18]. It was relatively easy to ®t and not only had excellent statistical results for our data, but also gave rise to interesting biological interpretations of its parameters. We discuss this model in detail in the next section. 5. Localization of the model After choosing it from considerations of how well all the models ®tted individual plot data, we ®tted [18] (see Table 4) using the combined data from all 31 plots in Data Set 1: H ˆ 9:67 ‡ 19:43f1 ÿ eÿ0:0323…Dÿ7:5† g

(6)

At this point we are interested in knowing if the residuals from this model were different across stands. An analysis of variance with PROC GLM de®nitely showed this to be the case. Thus, signi®cant improvement in prediction precision may be achieved by determining unique properties of each sample plot and relating model parameters to these properties. One of the reasons we like model [18] is that the three parameters are all biologically explainable. This fact also has a bearing on our ability to propose sensible parameter prediction relationships. The parameter a is Hmin, the height corresponding to the

324

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

minimum diameter (Dmin) of the stand. The sum b0ˆa‡b is the asymptotic or maximum possible height. Thus, parameter b ˆ b0 ÿ a ˆ Hmax ÿ Hmin ˆ Range is the range in heights. Parameter c is the rate-ofchange parameter (Rate). A large c value means the trees in such a stand tend to approach the asymptotic height quicker with respect to unit changes in DBH and, therefore, have greater height to DBH ratios than is the case in a stand with a smaller c value. Different stand structures with different species mixes will usually have different asymptotic heights and respective rates. Another interesting property of model [18] is that, for a given set of data, the parameters b0 and c are invariant to the Dmin term. That is to say, Dmin may be changed to, e.g. Dmin , as follows H  ˆ a ‡ b f1 ÿ eÿc



…DÿDmin †

g;

and b0 and c remain the same {i.e. a*‡b*ˆb0 and c*ˆc). Indeed, model [18] can be written as 

H  ˆ Hmin ‡ Rangef1 ÿ eÿRate…DÿDmin † g; where Hmin is the stand height corresponding to the minimum DBH of the stand, `Range' the asymptotic maximum minus Hmin, and `Rate' controls the rate of change in height in relation to a change in DBH. In this sense, it can be thought of as an index of stand structure. After determining our desired choice of model and con®rming that parameters should vary with stand characteristics, we set about developing parameter prediction relationships with stand summary variables that can be derived from DBH measurements. We elected not to consider the minimum and maximum DBH values as variables since these statistics are too strongly in¯uenced by outlying observations in the

 0:95 † of the DBH the average DBH of the largest 5% …D values on a plot as indicators of small and large diameters. Other variables considered include mean DBH, basal area, trees per ha, and standard deviation in DBH. All of these stand characteristics can easily be determined with forest inventory DBH measurements taken on ®xed-area plots. So, if the parameters of the height±diameter model have some kind of relationship with these stand characteristics, then the parameters may be predicted to obtain a localized model. After plotting the estimated parameters of [18] from individual plots in Data Set 1 over the above potential predictor variables and running a stepwise screening with PROC REG, it became apparent that the following linear models would be appropriate:  0:10 † ‡ "a ; a ˆ  ln…D

"a  N…0; 2a †

 0:95 ÿ D  0:10 † ‡ 2 ln …D  0:95 ÿ D  0:10 † ln …b† ˆ 0 ‡ 1 …D ‡ "b ;

"b  N…0; 2b †

(8)

 0:95 ÿ D0:10 †=bŠ=…D  0:95 ÿ D  0:10 † ‡ "c ; c ˆ 0 ln ‰…D "c  N…0; 2c †

(9)

Models (7) through (9) were ®tted using the individual parameter estimates (i.e. ^a; ^b and ^c from ®tting [18] (see Table 4) to each of the 31 plots in Data Set 1 as the response variables. All three regressions were signi®cant (p<0.0001) and all ®ve parameter estimates were signi®cantly different from zero (p<0.05): Parameter  0 1 2 

Estimate 4:7302 ÿ7:5840 ÿ0:0542 3:3778 1:7887

Standard error 0:0563 3:4364 0:0230 1:1760 0:0592

t-statistic value 84:009 ÿ2:207 ÿ2:358 2:872 30:224

Prob > jtj 0:0001 0:0357 0:0256 0:0077 0:0001 (10)

These parameter estimates were then used as starting values to ®t the model:

Hij ˆ  ln …D0:10i † ‡ exp… 0 ‡ 1 …D0:95i ÿ D0:10i † ‡ 2 ln …D0:95i ÿ D0:10i †† (  …Dij ÿD0:1i =…D0:95i ÿD0:10i † ) D0:95i ÿ D0:10i  1ÿ 0 ‡ 1 …D0:95i ÿ D0:10i ‡ 2 ln …D0:95i ÿ D0:10i † data. Instead of these two statistics, we elected to  0:10 † and use the average DBH of the smallest 10% …D

(7)

(11)

where Hijˆheight (m) of jth tree on ith plot, DIJ is the DBH (cm) of jth tree on ith plot, D0.10i the average of

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

325

Table 8 Parameters from the parameter prediction equations and prediction comparison with the overall average model for the height±DBH model of tropical stands on Hainan Island in southern China. Comparison based on 36 plots chosen at random (Data Set 2). Mean e0 is the average residual with the general model and Mean e1 and Mean e2 the average residuals when recovering the parameters with Eqs. (7)±(9). MSE0, MSE1 and MSE2 are the average mean squared errors for the same three respective approaches Plot

Mean e0

Mean e1

Mean e2

MES0

MES1

MES2

Sign a

1 3 4 5 7 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Average

ÿ4.092 ÿ2.383 ÿ2.953 ÿ3.498 ÿ1.889 ÿ3.092 ÿ3.096 ÿ2.318 ÿ1.958 ÿ4.035 ÿ3.235 ÿ3.005 ÿ3.308 ÿ3.262 ÿ3.403 ÿ2.360 ÿ2.727 ÿ3.199 ÿ2.979 ÿ2.872 ÿ3.482 ÿ2.042 ÿ1.776 ÿ2.915 ÿ2.378 ÿ2.382 ÿ3.199 ÿ3.279 ÿ2.270 ÿ2.345 ÿ2.961 ÿ4.060 ÿ1.547 ÿ1.908 ÿ3.258 ÿ2.895 ÿ2.84

ÿ1.4668 0.3549 ÿ0.2196 ÿ0.9764 0.5818 ÿ0.9676 ÿ0.8430 0.1625 0.6076 ÿ1.2985 ÿ0.7425 ÿ0.5448 ÿ0.6441 ÿ0.7774 ÿ0.9575 0.0961 ÿ0.0814 ÿ0.8384 ÿ0.3213 ÿ0.3823 ÿ0.9734 0.3751 0.4357 ÿ0.4242 0.2896 0.2720 ÿ0.8262 ÿ0.9326 0.3684 0.4014 ÿ0.4253 ÿ1.8077 1.2013 0.7923 ÿ0.8624 ÿ0.3605 ÿ0.326

ÿ0.84385 0.51499 ÿ0.02389 ÿ1.23017 0.78132 ÿ1.03020 ÿ0.95193 0.16089 1.23561 ÿ1.17502 ÿ0.00232 ÿ0.75214 ÿ0.35433 ÿ1.03337 ÿ1.26335 0.30399 0.42077 ÿ0.63878 0.00017 ÿ0.20027 ÿ0.45547 0.36510 0.09725 ÿ0.30997 0.61243 0.64817 ÿ1.18369 ÿ0.42885 0.82801 0.56779 ÿ0.14281 ÿ2.19631 1.33466 1.23132 ÿ0.70463 ÿ0.43584 ÿ0.174

19.451 7.088 10.047 16.941 7.414 13.418 14.114 8.662 6.496 17.721 13.165 11.721 14.325 13.808 14.800 11.541 11.513 15.053 10.978 13.897 16.238 7.412 6.998 12.093 8.875 8.834 13.980 14.206 7.156 7.674 11.501 21.928 5.833 8.142 12.928 11.752 11.88

2.204 0.950 1.022 9.644 4.673 6.308 7.190 3.271 4.019 2.213 1.667 5.125 4.418 5.869 8.202 5.780 4.556 5.201 1.580 5.754 3.720 4.128 4.728 3.489 2.157 2.180 6.781 2.863 2.340 2.290 2.239 13.213 6.028 7.946 2.923 4.050 4.462

4.997 1.626 1.373 6.640 4.321 5.934 6.485 3.267 3.110 3.237 3.374 3.741 3.941 4.244 5.226 6.188 4.043 5.746 2.396 5.966 5.164 4.096 4.261 4.189 3.562 3.551 4.829 4.764 2.221 2.472 3.284 9.898 5.000 5.042 3.737 3.739 4.324

ÿ ÿ ÿ ‡ ‡ ‡ ‡ ‡ ‡ ÿ ÿ ‡ ‡ ‡ ‡ ÿ ‡ ÿ ÿ ÿ ÿ ‡ ‡ ÿ ÿ ÿ ‡ ÿ ‡ ÿ ÿ ‡ ‡ ‡ ÿ ‡

Pooled

CSS b

SSE0 c

SSE1 c

SSE2 c

42096.53

35061.41 0.1671

13158.55 0.6874

12766.58 0.6967

Sum square R squared a

Sign ``‡'' means MSE2
the smallest 10% of the DBH values on ith plot, and D0.95i the average of the largest 5% of the DBH values on ith plot. Eq. (11) was derived by substituting the

expressions for a, b, and c given in Eqs. (7)±(9) into [18] and doing some minor algebraic rearranging. The Marquardt algorithm converged and all parameter

326

Z. Fang, R.L. Bailey / Forest Ecology and Management 110 (1998) 315±327

estimates had asymptotic 95% con®dence intervals that did not overlap zero:

Parameter  0 1 2 

Estimate 4:7267 ÿ18:4769 ÿ0:1124 6:9014 1:8984

that the parameters of [18] are easily related to summary values calculated from diameter measurements

Asymptotic 95% confidence interval…Lower† 4:6803 ÿ21:0706 ÿ0:1288 6:0252 1:8477

6. Model testing The stand structures and species composition represented in Data Set 2 are quit different from those in Data Set 1 which was used for model development (Tables 1±3). When we used the general Eq. (6) to predict the total tree heights of the stands in Data Set 2, it was no surprise to get considerable bias and large prediction errors (Table 8). When we subsequently tested Eq. (11) with Data Set 2 using the predicted values for the parameters based on linear models (i.e. Eq. (10) above), average bias decreased by 86%, average precision as measured by mean-squared error increased by 60%, and the R2pooled increased from 16.7 to 66.6% (Table 8). When the parameter estimates from the nonlinear least-squares ®tting process (i.e. Eq. (12) above) were used, an additional reduction in average bias of 6.6% was achieved and the R2pooled was increased to 71.5%. 7. Discussion and conclusions Most of the functions we selected (Table 4) performed well in describing the height±diameter relationships for the tropical forests on Hainan Island. For two-parameter models, the linear model [2] and nonlinear model [13] are recommended for their easy of ®tting and relatively good precision in prediction. For three parameter functions, model [18] is recommended both for its good behavior in ®tting efforts and the biological interpretations possible for its parameters. We do not recommend any of the fourparameter models. They all seem to be over-parameterized, thus resulting in instability when nonlinear least squares algorithms are used to ®t them. We found

Asymptotic 95% confidence interval…Upper† 4:7731 ÿ15:8832 ÿ0:0959 7:7776 1:9490

(12)

in tropical forest stands. These relationships were incorporated into equation [18] and a general model ®tted. This model may be used with inventory data on diameters to predict tree heights and, subsequently, to predict tree volumes with volume tables accessed by dbh and total height. Parameter c of equation [18] directly in¯uences the rate of change in total height with respect to a change in DBH for a given minimum and range in height. If we suppose that stands with similar species composition will have similar minimum and range values for height at equivalent periods in their development, the differences in c from stand-to-stand may re¯ect differences in stand structure due to site productivity differences. In this sense, it may be possible to use the rate parameter, c, to evaluate site productivity. A `good' site may well be expected to have a higher value for this rate parameter, all other factors being equal. Although in the realm of conjecture and needing a thorough examination with more detailed data, the potential for interpretation of the rate parameter in this way may hold promise.

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