1

Fig. 1. The location of a) the study area – Loutou forest farm in b) Yueyang of Hunan Province, c) southern China. View larger in new window/tab.

Table 1. The information of the measured variables for the 16 sample plots (Note: DBH: diameter at breat height; H: tree height; Cg: Cyclobalanopsis glauca; Rs: Rhododendron simsii; Cs: Castanopsis sclerophylla; Ce: Castanopsis eyriei; Lc: Loropetalum chinense; Cl: Cunninghamia lanceolate; Af: Alniphyllum fortune; Ca: Canarium album; Mr: Myrica rubra; Ss: Symplocos sumuntia).
Sample
plot number 		Altitude
(m) 		Slope
(degree) 		Slope
aspect 		Slope
position 		Crown
density 		Tree composition Number
of trees 		Mean DBH
(cm) 		Mean H
(m)
1 324 19 sunny medium 0.9 4Cg 3Rs 1Cs 1Ce 1Lc 63 15.4 11.1
2 340 17 sunny medium 0.7 4Cg 3Rs 2Lc 1Cl 54 12.9 9.5
3 312 22 sunny below 0.8 4Cg 3Rs 2Cl 1Ce 41 14.6 10.0
4 310 34 shady medium 0.8 4Cg 4Rs 1Lc 1Ca 54 12.5 9.3
5 330 20 sunny below 0.7 3Cg 2Ce 2Cl 2Rs 1Mr 30 14.9 9.7
6 312 32 sunny below 0.8 4Cg 3Rs 2Ce 1Lc 51 12.1 9.1
7 315 33 sunny below 0.7 3Cg 3Af 3Rs 1Cl 53 13.1 9.0
8 320 35 shady medium 0.8 4Cg 3Cs 3Lc 46 15.1 8.8
9 314 35 shady medium 0.7 4Cg 2Cs 2Rs 1Af 1Lc 29 14.3 11.3
10 318 32 shady upper 0.7 5Cg 3Rs 2Af 45 13.2 9.9
11 302 15 shady medium 0.7 4Cg 2Af 2Cl 2Lc 48 12.8 9.6
12 305 20 shady medium 0.7 5Cg 2Rs 2Af 1Cl 81 12.7 9.5
13 310 16 shady upper 0.6 5Cg 2Rs 2Cs 1Lc 50 12.3 12.5
14 325 27 sunny medium 0.8 6Cg 2Cs 1Rs 1Cl 52 14.6 9.5
15 308 30 sunny medium 0.6 3Cg 3Cs 2Rs 1Ce 1Af 40 9.5 9.6
16 285 20 shady below 0.8 4Cg 3Cl 1Ce 1Af 1Rs 100 12.1 10.3
Table 2. Characteristics of Cyclobalanopsis glauca tree height and diameter at breast height (DBH) measurements for the total, modeling and validation datasets based on sunny-gentle, sunny-steep, shady-gentle and shady-steep (Max: maximum; Min: minimum; SD: standard deviation).
Subsets Sample type Number Height (m) DBH (cm)
Mean Max Min SD Mean Max Min SD
Sunny-gentle Total 96 11.5 17.3 5.6 2.9 14.6 42.5 5.2 6.9
Modeling 77 11.5 17.3 5.6 3.0 14.7 42.5 5.2 7.3
Validation 19 11.6 16.3 6.1 2.8 14.3 22.7 7.3 5.1
Sunny-s teep Total 86 10.3 15.4 6.5 2.2 12.9 26.0 5.0 5.1
Modeling 69 10.2 15.4 6.5 2.3 13.0 26.0 5.0 5.4
Validation 17 10.5 14.4 7.8 1.8 12.4 18.6 6.1 4.1
Shady-gentle Total 139 10.4 14.9 3.5 2.5 12.8 29.0 5.1 4.9
Modeling 111 10.4 14.8 3.5 2.5 12.9 27.2 5.1 4.9
Validation 28 10.4 14.9 5.8 2.5 12.0 29.0 5.8 4.7
Shady- steep Total 105 10.6 15.0 6.4 2.0 15.2 41.6 5.1 7.0
Modeling 84 10.7 14.7 6.4 1.9 14.8 33.4 5.1 6.3
Validation 21 10.2 15.0 6.7 2.4 16.4 41.6 5.1 9.4
Table 3. The basic models used for comparison of modeling the relationship of tree height (H) with diameter at breast height (DBH) (Note: D is DBH; a, b, c and d are model parameters; M#: model numbers).
M# Model References M# Model References
M1 Linear M7 Yoshida (1928)
M2 Log M8 Schumacher (1939)
M3 Exponential M9 Richards (1959)
M4 Hyperbolic curve M10 Huang and Titus (1992)
M5 Quadratic M11 Wykoff et al. (1982)
M6 Gompertz (1825)
Table 4. The results of eleven basic models (M1 to M11) used for fitting the H-DBH relationship using the whole modeling dataset (H is tree height; DBH is diameter at breast height; a, b, c and d are model parameters; RMSE and R2 respectively are root mean square error and coefficient of determination between the observed and estimated values of H; and SSR is the sum of squared residuals).
Model parameter Model evaluation
M# a b c d R2 SSR RMSE
M1 0.321 6.241 0.624 972.654 1.511
M2 4.735 –1.323 0.683 821.522 1.389
M3 3.529 0.043 0.670 854.983 1.417
M4 19.897 216.848 10.988 0.685 816.203 1.384
M5 –0.0097 0.6517 3.8716 0.679 831.126 1.397
M6 15.02 1.471 0.116 0.686 813.143 1.382
M7 10.672 156.574 2.024 5.169 0.686 811.589 1.380
M8 16.936 5.552 0.676 839.931 1.404
M9 15.949 0.764 0.071 0.685 814.723 1.383
M10 0.874 0.235 0.682 821.891 1.389
M11 2.87 –6.62 0.681 826.471 1.393
Table 5. The results of the optimal models selected from the eleven basic models (M1 to M11) used for fitting the H-DBH relationship for each of the sub-datasets including sunny, shady, steep and gentle slope, and sunny-steep, sunny-gentle, shady-steep and shady-steep (a, b and c are model parameters; RMSE and R2 respectively are root mean square error and coefficient of determination between the observed and estimated values of H; SSR is the sum of squared residuals; and AIC is Akaike information criterion).
Dataset Model # Model parameter Model evaluation
a b c R2 SSR RMSE AIC
Sunny slope M6 16.704 1.416 0.094 0.695 394.29 1.472 146.7
Shady slope M6 13.945 1.597 0.140 0.702 379.41 1.247 113.7
Steep slope M2 3.922 0.451 0.714 232.83 1.104 85.4
Gentle slope M6 15.816 1.757 0.124 0.720 492.06 1.447 130.3
Sunny-steep M2 4.253 –0.249 0.622 154.55 1.341 54.4
Sunny-gentle M6 16.817 1.591 0.109 0.754 200.08 1.444 76.5
Shady-steep M3 4.094 0.359 0.818 73.03 0.834 –34.1
Shady-gentle M6 14.119 2.184 0.169 0.706 255.58 1.356 92.7
Table 6. The comparison of the results using the optimal models without and with the original Hegyi and improved Hegyi_I involved for sunny-steep, sunny-gentle, shady-steep and shady-gentle slope forests (a, b, c, f and g are model parameters; STE is the standard error of the estimated parameter; RMSE and R2 respectively are root mean square error and coefficient of determination between the observed and estimated values of H; SSR is the sum of squared residuals; and the symbol * implying the model parameter statistically is significantly different from zero at the significant level of 0.05). View in new window/tab.
Table 7. Significant difference test of the average absolute residuals from zero using the models without and with the competition indices Hegyi CI and Hegyi_I CI involved (δ1 is the average of the absolute residuals for the model without competition index, δ2 is the average of the absolute residuals for the model with the original Hegyi and δ3 is the average of the absolute residuals for the model with the improved Hegyi_I).
Sub-datasets Model δ1 δ2 δ3 δ1 vs. δ2 δ1 vs. δ3 δ2 vs. δ3
T value P value T value P value T value P value
Sunny-steep M2 1.10* 1.05* 0.98* 1.635 0.106 2.310 0.023 1.578 0.118
Sunny-gentle M6 1.18* 1.12* 1.09* 0.405 0.686 1.748 0.084 0.303 0.763
Shady-steep M3 0.65* 0.62* 0.57* 1.292 0.199 2.386 0.019 1.903 0.060
Shady-gentle M6 1.11* 1.09* 1.05* 0.676 0.500 1.906 0.059 1.495 0.137
2

Fig. 2. The scattered distributions and fitted curves of the observed height (H) against the observed diameter (DBH) for a) the whole dataset and different sub-datasets: b) sunny-steep; c) sunny-gentle; d) shady-steep; and e) shady-gentle.

Table 8. The improvement of tree H predictions using separate models at different slopes and aspects by comparison with the whole model based on root mean square error (RMSE) between the predicted and observed values of the validation dataset (W is the whole model, S1 is the separate models without competition index, S2 is the separate models with the original Hegyi and S3 is the separate models with the improved Hegyi_I.
Datasets RMSE Percentage of reduced error (%)
W S1 S2 S3 W vs. S1 W vs. S2 W vs. S3 S2 vs. S3
Sunny-steep slope 1.366 1.341 1.33 1.312 1.83 2.64 3.95 1.35
Sunny-gentle slope 1.635 1.444 1.443 1.348 11.68 11.74 17.55 6.58
Shady-steep slope 1.064 0.834 0.811 0.759 21.62 23.78 28.66 6.41
Shady-gentle slope 1.412 1.356 1.363 1.341 3.97 3.47 5.03 1.61