Chinese fir (

Chinese fir (^{3} as a dominant tree species (

A stem taper equation describes a mathematical relation between tree height and the stem diameter at that height. It is thus possible to calculate the stem diameter at any arbitrary height and conversely, to calculate the tree height for any arbitrary stem diameter. Consequently, the stem volume can be calculated for any log specification and a volume equation can be developed for classified product dimensions. Numerous and various mathematical taper functions have been developed in attempts to describe tree taper. Viewed from the structures of these equations, the different taper equations can be divided into three major categories: simple mathematical equations (

Some variables related to forest management have long been recognized such as planting density, fertilization, thinning and age (

The objective of this study was to develop a taper equation and quantify the effect of planting density on stem taper for Chinese fir in Southern China.

A total of 293 trees sampled from 12 plots of even-aged Chinese fir stands were used in the present study. The trees were taken from unthinned stands that were planted in 1981 for a density-effect study of Chinese fir in Fenyi County, Jiangxi Province, of southern China (

The total tree height (

Plant density (stems/ha) | Number of trees | Mean height (m) | S.D. | Mean dbh (cm) | S.D. |
---|---|---|---|---|---|

Fit data | |||||

B: 3,333 (2 × 1.5 m) | 30 | 13.49 | 2.42 | 12.52 | 2.61 |

C: 5,000 (2 × 1 m) | 48 | 12.85 | 2.29 | 11.03 | 2.48 |

D: 6,667 (1 × 1.5 m) | 54 | 12.02 | 1.90 | 10.26 | 2.10 |

E: 10,000 (1 × 1 m) | 51 | 11.83 | 2.86 | 9.62 | 2.09 |

Validation data | |||||

B: 3,333 (2 × 1.5 m) | 20 | 13.91 | 2.14 | 12.94 | 2.48 |

C: 5,000 (2 × 1 m) | 30 | 13.25 | 2.14 | 11.47 | 2.06 |

D: 6,667 (1 × 1.5 m) | 30 | 12.33 | 1.76 | 10.47 | 1.92 |

E: 10,000 (1 × 1 m) | 30 | 11.93 | 1.84 | 9.30 | 1.74 |

S.D. indicates standard deviation.

Five taper equations were analyzed in the present study, including two simple mathematical equations (

_{1.3}: stem radius at breast height, _{1}, _{2}, _{3} are parameters to be estimated.

Because

Based on dimensional analysis,

_{1} in _{4} is parameter.

_{4}, _{5}, _{6}, _{7}, _{8}, _{9} and

All the equations were fitted by the NLIN procedure in the SAS statistics program (

The model adjusted coefficient of determinations (

The accuracy of diameter predictions by these five taper equations was evaluated along the bole of Chinese fir trees using the validation data sets (_{8} of the Kozak equation had not significance at the 95% test level for maritime pine.

Model | _{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
_{7} |
_{8} |
_{9} |
S.E.E. | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1.6318 | −0.0294 | 0.9428 | 0.9649 | 0.6629 | ||||||||

9.5276 | −0.3199 | 0.6952 | 1.9540 | |||||||||

2.0307 | 0.9772 | 0.5335 | ||||||||||

0.9964 | 2.0294 | −0.0302 | 0.1051 | 0.9782 | 0.5224 | |||||||

1.2327 | 0.9968 | ns | 0.2408 | ns | 0.5006 | −0.4297 | ns | ns | ≈0 | 0.9785 | 0.5194 |

Based on the fitting and validation statistics, the Sharma Oderwald, Sharma/Zhang and modified Brink’s equations are suggested for use as taper equations for Chinese fir trees.

_{2} were significantly different between any two of the four planting densities (_{4} were significantly different between the four planting densities (_{1} were not significantly different between any two of the four planting densities (_{3} were found having no significance at the 95% level. These comparisons were made based on the confidence limits of the parameters obtained by nonlinear regressions.

Parameter | Planting density (stems/ha) | ||||
---|---|---|---|---|---|

3,333 | 5,000 | 6,667 | 10,000 | ||

_{1} |
2.1568(0.1928) | 1.5143(0.1079) | 1.6975(0.1371) | 1.4346(0.0984) | |

_{2} |
0.0011(0.0094) | −0.0436(0.0069) | −0.0243(0.0097) | −0.0392(0.0078) | |

_{3} |
0.6400(0.0781) | 1.2003(0.1722) | 0.8462(0.0963) | 1.1461(0.1409) | |

0.9761 | 0.9541 | 0.9653 | 0.9635 | ||

S.E.E. | 0.6455 | 0.8007 | 0.6236 | 0.6009 | |

_{1} |
2.0324(0.0015) | 2.0319(0.0013) | 2.0303(0.0010) | 2.0278(0.0011) | |

0.9815 | 0.9713 | 0.9810 | 0.9765 | ||

S.E.E. | 0.6161 | 0.6198 | 0.4565 | 0.4768 | |

_{1} |
0.9976(0.0077) | 0.9893(0.0085) | 0.9998(0.0064) | 0.9973(0.0074) | |

_{2} |
2.0280(0.0015) | 2.0326(0.0016) | 2.0293(0.0012) | 2.0273(0.0014) | |

_{3} |
ns | ns | −0.0712(0.0347) | ns | |

_{4} |
0.1625(0.0509) | 0.0557(0.0551) | 0.1561(0.0443) | 0.0741(0.0513) | |

0.9848 | 0.9712 | 0.9817 | 0.9769 | ||

S.E.E. | 0.4978 | 0.6180 | 0.4426 | 0.4724 |

To examine the effect of stand planting density on tree taper, the correlations between some parameters of _{1} of _{1} of ^{2}) between parameter _{1} of _{1} of _{1} of _{1}, _{2}, _{1} and _{2} are parameters to be estimated.

The test results of correlation coefficients showed that the parameters of

To examine the effect of stand planting density on tree taper in a Chinese fir plantation,

Parameter | Estimates of parameters | S.E.E. | ||
---|---|---|---|---|

_{1} |
2.1504 (0.5675) | 0.6627 | 0.9650 | |

_{2} |
−0.0306 (0.0292) | |||

_{2} |
−0.0282 (0.0040) | |||

_{3} |
0.9331 (0.0575) | |||

_{1} |
21.7021 (8.5436) | 0.5327 | 0.9773 | |

_{2} |
2.0266 (0.0017) | |||

_{1} |
0.9964 (0.0039) | 0.5219 | 0.9782 | |

_{2} |
2.0257 (0.0017) | |||

_{3} |
−0.0310 (0.0204) | |||

_{4} |
0.1058 (0.0259) | |||

_{5} |
19.6487 (8.3747) |

Relative height | Number | Bias (cm) | Absolute bias (cm) | ||||
---|---|---|---|---|---|---|---|

0.0 ≤ |
279 | 0.7662 | 0.0233 | 0.0829 | 0.7835 | 0.3214 | 0.3250 |

0.1 < |
205 | −0.0226 | −0.0185 | −0.0053 | 0.1164 | 0.1170 | 0.1228 |

0.2 < |
157 | −0.0215 | −0.0453 | −0.0379 | 0.1962 | 0.2054 | 0.2048 |

0.3 < |
151 | −0.0297 | −0.0984 | −0.0757 | 0.2683 | 0.2780 | 0.2745 |

0.4 < |
151 | 0.0414 | −0.0795 | −0.0237 | 0.3138 | 0.3089 | 0.3043 |

0.5 < |
153 | 0.0976 | −0.0598 | 0.0454 | 0.3910 | 0.3662 | 0.3656 |

0.6 < |
154 | 0.1041 | −0.0419 | 0.1170 | 0.4514 | 0.4164 | 0.4360 |

0.7 < |
160 | 0.0624 | −0.0564 | 0.1513 | 0.4992 | 0.4817 | 0.5161 |

0.8 < |
152 | −0.0556 | −0.2528 | −0.0177 | 0.4773 | 0.5498 | 0.5106 |

0.9 < |
139 | 0.2573 | −0.2779 | −0.0983 | 0.4725 | 0.5370 | 0.4635 |

Statistics | Equations with stand density variable | Equations without stand density variable | ||||
---|---|---|---|---|---|---|

M.D. | 0.1601 | −0.0752 | 0.0223 | 0.1612 | −0.0758 | 0.0226 |

M.A.D. | 0.4155 | 0.3452 | 0.3416 | 0.4157 | 0.3453 | 0.3418 |

S.E.E. | 0.6543 | 0.5007 | 0.4835 | 0.6548 | 0.5011 | 0.4838 |

The modified Brink’s equation mostly had larger and more positive bias at the butt and tip of tree stems than at mid-stem. However, the Sharma/Oderwald and Sharma/Zhang equations had larger bias at lower stem parts, and the Sharma/Oderwald equation had an almost negative bias along the boles excepting at the butt. The results show that the diameters at two ends of the stems of Chinese fir trees will be underestimated when using the modified Brink’s equation, and were mostly overestimated by the Sharma/Oderwald equation (

The maximum mean absolute bias of the modified Brink’s, Sharma/Oderwald and Sharma/Zhang equations with the stand density variable were 0.7662, 0.5498, and 0.5161 cm. Note that the modified Brink’s equation had relatively larger bias than the Sharma/Zhang and Sharma/Oderwald equations only because of the great bias at the butt of tree stems. Considering the variable-exponent taper equation’s theoretical property, the modified Brink’s and Sharma/Zhang equations were the most appropriate equations for describing tree taper of Chinese fir trees.

The effect of stand planting density was analyzed visually by generating tree profiles using

(A) showing the difference of tree profiles below 0.1 at four different densities.

Variable taper equations were developed for Chinese fir, the most important commercial tree species in southern China. The Sharma/Oderwald, Sharma/Zhang, and modified Brink’s equations are superior to the Pain/Boyer equation in terms of the fitting and validation statistics. The modified Brink’s equation only had lower prediction precision than the Sharma/Oderwald and Sharma/Zhang equations at the butt diameter. If the final choice must be made, the modified Brink’s equation and Sharma/Zhang equation are recommended for use as a taper equation for Chinese fir.

Correlation analysis results showed that stand planting density had an obvious effect on some parameters of taper equations. Therefore, the relationships between some parameters of the three selected equations and stand planting densities can be built by adopting some simple mathematical functions to examine the effect of stand planting density on tree taper.

The prediction precision of the three taper equations was compared with or without incorporation of the stand density variable. The M.D., A.M.D., and S.E.E. using for estimating diameters along the stems for the validation data sets showed that adding the stand density variable improved the evaluation efficacy of the taper equations for Chinese fir trees. The maximum mean absolute bias of the modified Brink’s and Sharma/Zhang equations with a stand density variable were all below 1.0 cm in the study area. The modelling difference of tree profiles among different stand densities mainly appeared below the 10% of total high.

Thanks go to Mr. Quang V. Cao in the Louisiana State University for his help with revising and suggestions. Special thanks go to two careful and warmhearted reviewers.

The authors declare there are no competing interests.

The following information was supplied regarding data availability:

The raw data has been supplied as