Table 1. Summary statistics of modelling data to test parameter recovery method of Weibull distribution and to develop Näslund’s height curve model for hybrid aspen (Populus tremula × P. tremuloides) from southern Finland. | |

Variables | Statistics, Mean ± SD (Min–Max) |

Dataset structure | |

No. of stands | 4 |

No. of plots | 48 |

No. of sample trees | 4481 |

No. of measurements by stand | 7–12 |

No. of observations at plot-level instances | 294 |

No. of observations at tree-level instances | 26 933 |

Stand characteristics | |

Stand age (AGE, years) | 5–20 |

Stand density (TPH, trees ha^{−1}) | 930 ± 423 (300–1600) |

Basal area (BA, m^{2} ha^{−1}) | 10.8 ± 9.0 (0.1–35.7) |

Diameter, arithmetic mean (D, cm) | 11.1 ± 5.6 (1.5–24.9) |

Diameter, basal area-weighted mean (DG, cm) | 12.0 ± 5.7 (2.1–26.8) |

Diameter, median (DM, cm) | 11.3 ± 5.8 (1.2–26.0) |

Diameter, basal area median (DGM, cm) | 12.1 ± 5.8 (1.9–26.6) |

Diameter, quadratic mean (DQ, cm) | 11.4 ± 5.6 (1.8–25.4) |

Height, basal area-weighted mean (HG, m) | 13.5 ± 6.6 (3.6–27.6) |

Dominant height (HDOM, m) | 15.1 ± 6.7 (4.0–29.8) |

Site index ^{a} (SI, m) | 25.5 ± 2.7 (17.5–29.9) |

Tree characteristics | |

Diameter at breast height (dbh, cm) | 10.8 ± 5.7 (0.1–31.8) |

Height (h, m) | 13.6 ± 6.8 (1.4–31.4) |

^{a}: Site index was computed based on Lee et al. (2021) with a base age of 20 years. |

Table 2. The goodness-of-fit statistics for the recovered Weibull functions using arithmetic mean (D), basal area weighted mean (DG), median (DM), and basal area median dbh (DGM) for resolving the parameters based on the data of hybrid aspen (Populus tremula × P. tremuloides) from southern Finland. The smaller the value of test criterion, the better the fit. The values are mean ± standard deviation with minimum-maximum in parentheses. | ||||

Variable | Kolmogorov-Smirnov quotient | Rejection cases of Kolmogorov-Smirnov at 10% level | The proportion of rejected cases to 294 total cases (%) | Error index by Reynolds et al. (1988) |

D | 0.535 ± 0.232 (0.054–1.478) | 15 | 5.1 | 23.29 ± 11.52 (0.67–56.43) |

DG | 0.507 ± 0.213 (0.134–1.425) | 8 | 2.7 | 22.75 ± 11.19 (2.67–60.31) |

DM | 1.463 ± 2.673 (0.175–29.528) | 101 | 34.4 | 50.73 ± 58.38 (4.83–470.71) |

DGM | 0.675 ± 0.741 (0.064–11.212) | 37 | 12.6 | 26.41 ± 13.11 (1.39–91.62) |

Note: Kolmogorov-Smirnov quotient is the supremum divided by the critical value. |

Table 3. Parameter estimates and fit statistics for Näslund’s height curve models to predict individual tree height of hybrid aspen (Populus tremula × P. tremuloides) in southern Finland. | |||||||

Model 1 | Model 2 | ||||||

Estimate | S.E. | Estimate | S.E. | ||||

Fixed effects | b_{0} | a_{0} | Intercept | −0.2531 | 0.0493 | 0.4180 | 0.0592 |

a_{1} | ln(DG) | 0.4124 | 0.0229 | 0.3788 | 0.0204 | ||

a_{2} | ln(HG) | −0.2723 | 0.0285 | −0.2585 | 0.0254 | ||

a_{3} | ln(AGE) | −0.1683 | 0.0263 | −0.2184 | 0.0227 | ||

a_{4} | ln(TPH) | −0.0658 | 0.0060 | ||||

b_{1} | a_{0} | Intercept | −0.4411 | 0.0093 | 2.3409 | 0.0420 | |

a_{1} | ln(DG) | 0.1277 | 0.0069 | −0.4653 | 0.0107 | ||

a_{2} | ln(HG) | −0.4804 | 0.0100 | −0.5922 | 0.0093 | ||

a_{3} | ln(AGE) | −0.0431 | 0.0082 | 0.0479 | 0.0071 | ||

a_{4} | ln(BA) | −0.0156 | 0.0012 | 0.3039 | 0.0044 | ||

a_{5} | ln(TPH) | −0.2880 | 0.0043 | ||||

Random effects | u_{0} | std(experiment) | 0.0851 | 0.0663 | |||

u_{1} | std(experiment) | 0.0097 | 0.0190 | ||||

Corr(u_{0}, u_{1}) | –0.413 | –0.998 | |||||

Residual | std(ε) | 0.2489 | 0.2719 | ||||

Variance function | Power | 0.4459 | 0.3769 | ||||

Fit statistics | AIC | 60 500.64 | 56 125.17 | ||||

BIC | 60 615.46 | 56 256.38 | |||||

–2logLik | 60 472.64 | 56 093.16 | |||||

AIC is the Akaike information criterion. BIC is the Bayesian information criterion. –2logLik is the –2 × log-likelihood value. Corr is the correlation between random-effect parameters. All the fixed-effect parameters were highly significant (P < 0.0001). |