Table 1. Statistics of stand-and tree-level variables used for modelling height-diameter allometry of Chinese fir in this study. | ||||||||||

Planting Density (PD) | BA (m^{2} ha^{–1}) | N (tree/ha) | HD (m) | DBH (cm) | H (m) | |||||

Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD | |

A (1667 trees/ha) | 33.03 | 19.84 | 1588.27 | 193.43 | 14.60 | 6.69 | 15.61 | 7.32 | 12.20 | 6.24 |

B (3333 trees/ha) | 36.99 | 20.63 | 3094.15 | 513.69 | 13.14 | 6.12 | 12.59 | 5.86 | 10.94 | 5.51 |

C (5000 trees/ha) | 40.47 | 21.95 | 4415.94 | 976.48 | 13.61 | 6.45 | 12.04 | 5.87 | 11.22 | 5.74 |

D (6667 trees/ha) | 44.58 | 24.36 | 5705.04 | 1412.55 | 12.77 | 6.28 | 11.21 | 5.36 | 10.66 | 5.34 |

E (10 000 trees/ha) | 42.21 | 19.97 | 7719.66 | 2794.09 | 12.27 | 5.94 | 9.87 | 5.13 | 9.41 | 4.87 |

BA: Stand basal area, N: Number of trees per ha; HD: stand dominant height; DBH: diameter at breast height; H: Tree height. Mean and SD calculated over the 17 instances of field sampling measurements taken from 1984–2010 (with each sampling instance occurring every year from 1984 to 1990, and every other year from 1992 to 2010). |

Table 2. Summary statistics of climate variables for the years 1984–2010 used for modelling height-diameter allometry. Values in parentheses are minimum and maximum values. | ||

Climate variable | Description | Mean |

MAT (°C) | Mean annual temperature | 18.96 (18.10, 19.80) |

MWMT (°C) | Mean warmest month temperature | 28.26 (26.50, 30.30) |

MCMT (°C) | Mean coldest month temperature | 8.34 (5.20, 10.20) |

AP (mm) | Annual precipitation | 1795.79 (1390.00, 2416.00) |

AHM | Annual heat-moisture index | 16.45 (11.90, 21.40) |

SMMT (°C) | Summer mean maximum temperature | 32.10 (30.30, 33.80) |

WMMT (°C) | Winter mean minimum temperature | 4.95 (2.50, 6.60) |

SMT (°C) | Spring (Mar.–May) mean temperature | 18.53 (16.90, 19.60) |

Table 3. Inference rules for determining if variables have an effect on tree height using the Bayesian model averaging (BMA) posterior probability. | |

Probability | Effect of x_{j} on tree height |

P(β_{j} ≠ 0 | y) < 0.5 | no effect |

0.5 ≤ P(β_{j} ≠ 0 | y) < 0.75 | weak effect |

0.75 ≤ P(β_{j} ≠ 0 | y) < 0.95 | positive effect |

P(β_{j} ≠ 0 | y) ≥ 0.95 | strong effect |

Table 4. The top five models selected and their posterior probabilities (post prob) of tree height-diameter allometry through Bayesian model averaging (BMA) and the model selected by stepwise approach (SR). The BMA model that has same variables as the SR model is in bold. | |||

BMA models | SR Model | ||

Models | Post prob | DBH, PD, HD, BA, MAT, MCMT, SMMT, WMMT | |

Model 1 | (0.454) | DBH, PD, HD, BA, MAT, MCMT, SMMT | |

Model 2 | (0.312) | DBH, PD, HD, BA, MAT, SMMT, WMMT | |

Model 3 | (0.127) | DBH, PD, HD, BA, MAT, MCMT, SMMT, WMMT | |

Model 4 | (0.082) | DBH, PD, HD, BA, MAT, MCMT, AHM, SMMT | |

Model 5 | (0.016) | DBH, PD, HD, BA, MCMT, SMMT | |

BA: Stand basal area, HD: stand dominant height; DBH: diameter at breast height; H: Tree height; PD: planting density; MAT: mean annual temperature; MCMT: mean coldest month temperature; SMMT: Summer mean maximum temperature; WMMT: Winter mean minimum temperature; AHM: Annual heat-moisture index. |

Table 5. Evaluation statistics of model prediction determined by Bayesian model averaging (BMA) and stepwise (SR) methods for modelling height-diameter allometry. | ||

Statistics | BMA | SR |

R^{2} | 0.9541 | 0.9540 |

MD | 0.0018 | –0.0078 |

MAD | 0.9042 | 0.9043 |

MD: mean difference; MAD: mean absolute difference. |

Table 6. The parameter estimates determined by Bayesian model averaging (BMA) and stepwise (SR) methods for modelling height-diameter allometry. | ||||||

Variable | SR | BMA | ||||

Mean | 95%CI | P-value | Mean | 95% CI | PP | |

Intercept | 0.599 | –0.177, 1.574 | >0.05 | 2.263 | 2.260, 2.266 | 1.000 |

DBH | 0.540 | 0.526, 0.664 | <0.01 | 0.540 | 0.527, 0.554 | 1.000 |

PD | 0.106 | 0.099, 0.114 | <0.01 | 0.106 | 0.099, 0.113 | 0.999 |

HD | 0.678 | 0.664, 0.695 | <0.01 | 0.678 | 0.661, 0.690 | 0.999 |

BA | –0.099 | –0.107, –0.091 | <0.01 | –0.099 | –0.107, –0.089 | 1.000 |

MAT | 0.563 | 0.341, 0.784 | <0.01 | 0.498 | 0.241, 0.648 | 0.991 |

MCMT | –0.059 | –0.099, –0.018 | <0.01 | –0.056 | –0.123, –0.055 | 0.714 |

AHM | - | - | >0.05 | 0.002 | 0.000, 0.032 | 0.098 |

SMMT | –0.989 | –1.250, –0.728 | <0.01 | –0.974 | –1.275, –0.670 | 1.000 |

WMMT | –0.037 | –0.064, –0.010 | <0.01 | –0.024 | –0.070, 0.000 | 0.413 |

BA: Stand basal area, HD: stand dominant height; DBH: diameter at breast height; PD: planting density; MAT: mean annual temperature; MCMT: mean coldest month temperature; SMMT: Summer mean maximum temperature; WMMT: Winter mean minimum temperature; AHM: Annual heat-moisture index; CI: confidence interval for SR, and credible interval for BMA. PP: Posterior probability from BMA and values in bold indicate that variables have a strong effect on tree height. |

Table 7. Summary of the scale of height-diameter allometry. | |||||

Scale | Study by Mcmahon and Kronauer (1976) | Study by Zhang et al. (2019) | This study | ||

Stress similarity | Elastic similarity | Geometric similarity | |||

0.50 | 0.66 | 1.00 | Close to 0.5 | 0.54 |