# Assessing the Effects of Sample Size on Parametrizing a Taper Curve Equation and the Resultant Stem-Volume Estimates

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Terrestrial Laser Scanning (TLS) Data Acquisition

^{2}were established in the study area (Figure 1). Locations of the field sample plots were selected based on canopy density and height information derived from airborne laser scanning data in order to cover the variation in these attributes within the study area. All trees with DBH ≥5 cm within the plots were measured using traditional field measurement practices [25] and a total of 2351 Scots pines were identified from 37 of these sample plots that were dominated by Scots pine. The basal-area weighted mean diameter varied between 14.8 and 32.0 cm in these 37 sample plots, development class from young-managed to regeneration-ready stands, and site type from mesic heath to xeric heath forest. In addition, TLS data were acquired in these plots using a Leica HDS6100 (Leica Geosystems AG, Heerbrugg, Switzerland) in the summer of 2014. The plots were scanned as they are (i.e., no removal of lower tree branches or undergrowth vegetation carried out). Point clouds were obtained from five locations (i.e., plot center and four quadrant directions) (Figure 2) with “High-density” mode providing the angle increment of 0.036° in both horizontal and vertical directions and point spacing of 15.7 mm at 25 m from the scan location. Artificial reference targets (i.e., white spheres with a radius of 198 mm) were used to co-register and combine separate point clouds into one, plot-specific point cloud with an average registration accuracy of 2.1 mm.

#### 2.2. Taper Curve Measurements from TLS Data

#### 2.3. Validation Data

#### 2.4. Modelling

_{l}is the diameter at a relative height of l from the ground, d

_{.2h}is the diameter at 20% relative height, and x is the relative distance from the tip of a tree.

#### 2.5. Evaluation of Results

_{ref}is the stem volume calculated based on the taper curves measured with the validation pines (i.e., destructive measurements) and V

_{pred}is the stem volume estimated with taper curve equation parametrized based on taper curves measured automatically from the TLS point clouds. The RMSE and mean difference were divided by the mean volume based on taper curves measured from validation pines and multiplied by 100 to obtain the relative RMSE and mean difference.

## 3. Results

^{3}when the sample size included at least 10 Scots pines corresponding to the relative mean RMSE of ~21% (Table 2). The standard deviation of absolute and relative RMSE followed a similar pattern and were ≤15.8 dm

^{3}and 5.6%, respectively, when sample size was ≥10 Scots pines. Also, the mean absolute and relative mean difference were −22 dm

^{3}and 9%, respectively, when the sample size was ≥10 Scots pines. However, the standard deviation of mean difference increased from 10.9 dm

^{3}to 14.3 dm

^{3}(from 4.4% to 5.8%) when the sample size decreased from 15 to 10 Scots pines.

^{3}(18.95%), which is in same magnitude than the RMSEs obtained from the parametrized taper curve equation based on the TLS data. The mean difference of 3.69 dm

^{3}(1.52%), however, was notably different when compared to the mean difference of stem-volume estimates obtained by parametrizing the taper curve equation with our TLS data. Taper curve based on the existing equation (i.e., coefficients estimated in [29]) produced smaller diameters throughout the relative heights, except for the lowest relative heights (Figure 9).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bettinger, P.; Boston, K.; Siry, J.P.; Grebner, D.L. Forest Management and Planning; Academic Press: New York, NY, USA, 2016. [Google Scholar]
- Kangas, A.; Maltamo, M. Forest Inventory: Methodology and Applications; Springer Science & Business Media: New York, NY, USA, 2006; Volume 10. [Google Scholar]
- Lehtonen, A.; Mäkipää, R.; Heikkinen, J.; Sievänen, R.; Liski, J. Biomass expansion factors (BEFs) for Scots pine, Norway spruce and birch according to stand age for boreal forests. For. Ecol. Manag.
**2004**, 188, 211–224. [Google Scholar] [CrossRef] - Brown, S. Measuring carbon in forests: Current status and future challenges. Environ. Pollut.
**2002**, 116, 363–372. [Google Scholar] [CrossRef] - Petersson, H.; Holm, S.; Ståhl, G.; Alger, D.; Fridman, J.; Lehtonen, A.; Lundström, A.; Mäkipää, R. Individual tree biomass equations or biomass expansion factors for assessment of carbon stock changes in living biomass—A comparative study. For. Ecol. Manag.
**2012**, 270, 78–84. [Google Scholar] [CrossRef] - Albaugh, T.J.; Bergh, J.; Lundmark, T.; Nilsson, U.; Stape, J.L.; Allen, H.L.; Linder, S. Do biological expansion factors adequately estimate stand-scale aboveground component biomass for Norway spruce? For. Ecol. Manag.
**2009**, 258, 2628–2637. [Google Scholar] [CrossRef] - Whitehead, D. Forests as carbon sinks—Benefits and consequences. Tree Physiol.
**2011**, 31, 893–902. [Google Scholar] [CrossRef] [PubMed] - Chave, J.R.; Andalo, C.; Brown, S.; Cairns, M.A.; Chambers, J.; Eamus, D.; Fölster, H.; Fromard, F.; Higuchi, N.; Kira, T. Tree allometry and improved estimation of carbon stocks and balance in tropical forests. Oecologia
**2005**, 145, 87–99. [Google Scholar] [CrossRef] [PubMed] - Duncanson, L.I.; Dubayah, R.O.; Enquist, B.J. Assessing the general patterns of forest structure: Quantifying tree and forest allometric scaling relationships in the United States. Glob. Ecol. Biogeogr.
**2015**, 24, 1465–1475. [Google Scholar] [CrossRef] - Henry, M.; Picard, N.; Trotta, C.; Manlay, R.; Valentini, R.; Bernoux, M.; Saint André, L. Estimating tree biomass of sub-Saharan African forests: A review of available allometric equations. Silva Fenn.
**2011**, 45, 477–569. [Google Scholar] [CrossRef] - Melson, S.L.; Harmon, M.E.; Fried, J.S.; Domingo, J.B. Estimates of live-tree carbon stores in the Pacific Northwest are sensitive to model selection. Carbon Balance Manag.
**2011**, 6, 2. [Google Scholar] [CrossRef] - Picard, N.; Saint-André, L.; Henry, M. Manual for Building Tree Volume and Biomass Allometric Equations: From Field Measurement to Prediction; Food and Agricultural Organization of the United Nations: Rome, Italy, 2012. [Google Scholar]
- Zianis, D.; Muukkonen, P.; Mäkipää, R.; Mencuccini, M. Biomass and Stem Volume Equations for Tree Species in Europe; FI: Vantaa, Finland, 2005. [Google Scholar]
- Dassot, M.; Constant, T.; Fournier, M. The use of terrestrial LiDAR technology in forest science: Application fields, benefits and challenges. Ann. For. Sci.
**2011**, 68, 959–974. [Google Scholar] [CrossRef] - Liang, X.; Kankare, V.; Hyyppä, J.; Wang, Y.; Kukko, A.; Haggrén, H.; Yu, X.; Kaartinen, H.; Jaakkola, A.; Guan, F. Terrestrial laser scanning in forest inventories. ISPRS J. Photogramm. Remote Sens.
**2016**, 115, 63–77. [Google Scholar] [CrossRef] - Saarinen, N.; Kankare, V.; Vastaranta, M.; Luoma, V.; Pyörälä, J.; Tanhuanpää, T.; Liang, X.; Kaartinen, H.; Kukko, A.; Jaakkola, A. Feasibility of Terrestrial laser scanning for collecting stem volume information from single trees. ISPRS J. Photogramm. Remote Sens.
**2017**, 123, 140–158. [Google Scholar] [CrossRef] - Disney, M.I.; Boni Vicari, M.; Burt, A.; Calders, K.; Lewis, S.L.; Raumonen, P.; Wilkes, P. Weighing trees with lasers: Advances, challenges and opportunities. Interface Focus
**2018**, 8, 20170048. [Google Scholar] [CrossRef] [PubMed] - Liang, X.; Hyyppä, J.; Kaartinen, H.; Lehtomäki, M.; Pyörälä, J.; Pfeifer, N.; Holopainen, M.; Brolly, G.; Francesco, P.; Hackenberg, J. International benchmarking of terrestrial laser scanning approaches for forest inventories. ISPRS J. Photogramm. Remote Sens.
**2018**, 144, 137–179. [Google Scholar] [CrossRef] - Henning, J.G.; Radtke, P.J. Detailed stem measurements of standing trees from ground-based scanning lidar. For. Sci.
**2006**, 52, 67–80. [Google Scholar] - Maas, H.G.; Bienert, A.; Scheller, S.; Keane, E. Automatic forest inventory parameter determination from terrestrial laser scanner data. Int. J. Remote Sens.
**2008**, 29, 1579–1593. [Google Scholar] [CrossRef] - Thies*, M.; Pfeifer, N.; Winterhalder, D.; Gorte, B.G. Three-dimensional reconstruction of stems for assessment of taper, sweep and lean based on laser scanning of standing trees. Scand. J. For. Res.
**2004**, 19, 571–581. [Google Scholar] [CrossRef] - Liang, X.; Kankare, V.; Yu, X.; Hyyppa, J.; Holopainen, M. Automated Stem Curve Measurement Using Terrestrial Laser Scanning. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 1739–1748. [Google Scholar] [CrossRef] - Sun, Y.; Liang, X.; Liang, Z.; Welham, C.; Li, W. Deriving merchantable volume in poplar through a localized tapering function from non-destructive terrestrial laser scanning. Forests
**2016**, 7, 87. [Google Scholar] [CrossRef] - Päivinen, R.; Nousiainen, M.; Korhonen, K.T. Puustotunnusten mittaamisen luotettavuus. Folia For.
**1992**, 787, 24. [Google Scholar] - Luoma, V.; Saarinen, N.; Wulder, M.; White, J.; Vastaranta, M.; Holopainen, M.; Hyyppä, J. Assessing precision in conventional field measurements of individual tree attributes. Forests
**2017**, 8, 38. [Google Scholar] [CrossRef] - Abegg, M.; Kükenbrink, D.; Zell, J.; Schaepman, M.; Morsdorf, F. Terrestrial laser scanning for forest Inventories—Tree diameter distribution and scanner location impact on occlusion. Forests
**2017**, 8, 184. [Google Scholar] [CrossRef] - Forsman, M.; Börlin, N.; Olofsson, K.; Reese, H.; Holmgren, J. Bias of cylinder diameter estimation from ground-based laser scanners with different beam widths: A simulation study. ISPRS J. Photogramm. Remote Sens.
**2018**, 135, 84–92. [Google Scholar] [CrossRef] - Kankare, V.; Holopainen, M.; Vastaranta, M.; Puttonen, E.; Yu, X.; Hyyppä, J.; Vaaja, M.; Hyyppä, H.; Alho, P. Individual tree biomass estimation using terrestrial laser scanning. ISPRS J. Photogramm. Remote Sens.
**2013**, 75, 64–75. [Google Scholar] [CrossRef] - Laasasenaho, J. Taper Curve and Volume Functions for Pine, Spruce and Birch; Metsäntutkimuslaitos: Vantaa, Finland, 1982. [Google Scholar]
- Bruce, D.; Curtis, R.O.; Vancoevering, C. Development of a System of Taper and Volume Tables for Red Alder. For. Sci.
**1968**, 14, 339–350. [Google Scholar] [CrossRef] - Pienaar, L.V.; Turnbull, K.J. The Chapman-Richards Generalization of Von Bertalanffy’s Growth Model for Basal Area Growth and Yield in Even–Aged Stands. For. Sci.
**1973**, 19, 2–22. [Google Scholar] [CrossRef] - Shinozaki, K.; Yoda, K.; Hozumi, K.; Kira, T. A quantitative analysis of plant form-the pipe model theory: I. Basic analyses. Jpn. J. Ecol.
**1964**, 14, 97–105. [Google Scholar] - Fehrmann, L.; Kleinn, C. General considerations about the use of allometric equations for biomass estimation on the example of Norway spruce in central Europe. For. Ecol. Manag.
**2006**, 236, 412–421. [Google Scholar] [CrossRef] - Kilkki, P.; Varmola, M. Taper Curve Models for Scots Pine and Their Applications; Suomen metsätieteellinen seura: Helsinki, Finland, 1981. [Google Scholar]
- Lappi, J. Mixed Linear Models for Analyzing and Predicting Stem Form Variation of SCOTS Pine; Metsäntutkimuslaitos: Vantaa, Finland, 1986. [Google Scholar]
- Diéguez-Aranda, U.; Castedo-Dorado, F.; Álvarez-González, J.G.; Rojo, A. Compatible taper function for Scots pine plantations in northwestern Spain. Can. J. For. Res.
**2006**, 36, 1190–1205. [Google Scholar] [CrossRef] - Subedi, N.; Sharma, M.; Parton, J. Effects of sample size and tree selection criteria on the performance of taper equations. Scand. J. For. Res.
**2011**, 26, 555–567. [Google Scholar] [CrossRef] - David, H.C.; Miranda, R.O.V.; Welker, J.; Fiorentin, L.D.; Ebling, Â.A.; Silva, P.H.B.M.d. Strategies for stem measurement sampling: A statistical approach of modelling individual tree volume. Cerne
**2016**, 22, 249–260. [Google Scholar] [CrossRef] - Kitikidou, K.; Chatzilazarou, G. Estimating the sample size for fitting taper equations. J. For. Sci.
**2008**, 54, 176–182. [Google Scholar] [CrossRef] [Green Version] - Kuželka, K.; Marušák, R. Use of nonparametric regression methods for developing a local stem form model. J. For. Sci.
**2014**, 60, 464–471. [Google Scholar] [CrossRef] [Green Version] - Westfall, J.A.; Scott, C.T. Taper models for commercial tree species in the Northeastern United States. For. Sci.
**2010**, 56, 515–528. [Google Scholar] - Vauhkonen, J.; Ene, L.; Gupta, S.; Heinzel, J.; Holmgren, J.; Pitkänen, J.; Solberg, S.; Wang, Y.; Weinacker, H.; Hauglin, K.M. Comparative testing of single-tree detection algorithms under different types of forest. Forestry
**2011**, 85, 27–40. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Hyyppä, J.; Liang, X.; Kaartinen, H.; Yu, X.; Lindberg, E.; Holmgren, J.; Qin, Y.; Mallet, C.; Ferraz, A. International benchmarking of the individual tree detection methods for modeling 3-D canopy structure for silviculture and forest ecology using airborne laser scanning. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 5011–5027. [Google Scholar] [CrossRef] - Axelsson, P. DEM generation from laser scanner data using adaptive TIN models. Int. Arch. Photogramm. Remote Sens.
**2000**, 33, 110–117. [Google Scholar] - Saarinen, N.; Vastaranta, M.; Kankare, V.; Tanhuanpää, T.; Holopainen, M.; Hyyppä, J.; Hyyppä, H. Urban-Tree-Attribute Update Using Multisource Single-Tree Inventory. Forests
**2014**, 5, 1032–1052. [Google Scholar] [CrossRef] [Green Version] - Vastaranta, M.; Saarinen, N.; Kankare, V.; Holopainen, M.; Kaartinen, H.; Hyyppä, J.; Hyyppä, H. Multisource Single-Tree Inventory in the Prediction of Tree Quality Variables and Logging Recoveries. Remote Sens.
**2014**, 6, 3475–3491. [Google Scholar] [CrossRef] [Green Version] - Pyörälä, J.; Liang, X.; Vastaranta, M.; Saarinen, N.; Kankare, V.; Wang, Y.; Holopainen, M.; Hyyppä, J. Quantitative assessment of Scots pine (Pinus sylvestris L.) whorl structure in a forest environment using terrestrial laser scanning. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2018**, 11, 3598–3607. [Google Scholar] [CrossRef] - Yu, X.; Liang, X.; Hyyppä, J.; Kankare, V.; Vastaranta, M.; Holopainen, M. Stem biomass estimation based on stem reconstruction from terrestrial laser scanning point clouds. Remote Sens. Lett.
**2013**, 4, 344–353. [Google Scholar] [CrossRef] - Green, P.J.; Silverman, B.W. Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach; Chapman and Hall/CRC: Boca Raton, FL, USA, 1993. [Google Scholar]
- Grosenbaugh, L. Improved cubic volume computation. J. For.
**1948**, 46, 299–301. [Google Scholar] - Stovall, A.E.; Anderson-Teixeira, K.J.; Shugart, H.H. Assessing terrestrial laser scanning for developing non-destructive biomass allometry. For. Ecol. Manag.
**2018**, 427, 217–229. [Google Scholar] [CrossRef] - Cormier, K.L.; Reich, R.M.; Czaplewski, R.L.; Bechtold, W.A. Evaluation of weighted regression and sample size in developing a taper model for loblolly pine. For. Ecol. Manag.
**1992**, 53, 65–76. [Google Scholar] [CrossRef] [Green Version] - Pyörälä, J.; Liang, X.; Saarinen, N.; Kankare, V.; Wang, Y.; Holopainen, M.; Hyyppä, J.; Vastaranta, M. Assessing branching structure for biomass and wood quality estimation using terrestrial laser scanning point clouds. Can. J. Remote Sens.
**2018**, 44, 462–475. [Google Scholar] [CrossRef] [Green Version] - Newton, P.F.; Sharma, M. Evaluation of sampling design on taper equation performance in plantation-grown Pinus banksiana. Scand. J. For. Res.
**2008**, 23, 358–370. [Google Scholar] [CrossRef] - Tabacchi, G.; Di Cosmo, L.; Gasparini, P. Aboveground tree volume and phytomass prediction equations for forest species in Italy. Eur. J. For. Res.
**2011**, 130, 911–934. [Google Scholar] [CrossRef] - Luoma, V.; Saarinen, N.; Kankare, V.; Tanhuanpää, T.; Kaartinen, H.; Kukko, A.; Holopainen, M.; Hyyppä, J.; Vastaranta, M. Examining Changes in Stem Taper and Volume Growth with Two-Date 3D Point Clouds. Forests
**2019**, 10, 382. [Google Scholar] [CrossRef]

**Figure 1.**Spatial distribution of 37 sample plots in the study site. The study site is illustrated by aerial imagery (red-green-blue, RGB) obtained in the summer of 2014.

**Figure 2.**Scan setup with multiple scan positions for terrestrial laser scanning data acquisition from 32 × 32 m sample plots.

**Figure 3.**Filtered canopy height model based on terrestrial laser scanning point cloud of a plot with segments and detected trees to illustrate the quality of the point cloud near the plot center (left) and at the border (right) in characterizing individual trees.

**Figure 4.**Relationship between diameter-at-breast height (DBH) and height of all Scots pines in 37 sample plots within the study area, Scots pines with taper curve measurements from terrestrial laser scanning and used for parametrizing the taper curve (i.e., modelling pines), and Scots pines measured destructively.

**Figure 5.**Distributions of diameter at breast height (DBH) classes of all Scots pines in 37 sample plots within the study area (grey), Scots pines used for parametrizing the taper curve (i.e., modelling pines) (blue), and Scots pines used for validating the reliability of parametrized taper curve equations (red). Additionally, distribution of Scots pines with TLS-based taper curves randomly sampled 100 times (without replacement) and their variation (error bars display standard deviation).

**Figure 6.**Range in coefficients of the taper curve equation developed with various sample sizes of Scots pines. (

**A**) Coefficient b1; (

**B**) Coefficient b2; (

**C**) Coefficient b3; (

**D**) Coefficient b4; (

**E**) Coefficient b5; (

**F**) Coefficient b6; (

**G**) Coefficient b7; (

**H**) Coefficient b8.

**Figure 7.**Box plots of relative root mean square error (RMSE) and mean difference of volume estimates based on parametrized taper curves obtained with samples of varying sizes that were randomly selected by 100 times. The box represents first and third quartile (i.e., 25th and 75th percentile), black line the median, and whiskers present the minimum and maximum of the relative RMSE and mean difference of stem-volume estimates.

**Figure 8.**The effect of sample size on relative root mean square error (RMSE; brown) and relative mean difference (blue) of stem-volume estimates. Error bars present the standard deviation around the mean that is also displayed.

**Figure 9.**Errors in estimated taper curves based on parametrized equation using all (green) as well as randomly selected 96 (blue) and one (brown) Scots pines for modelling. Additionally, difference in taper curve based on the coefficients by Laasasenaho (1982) (i.e., existing equation, black dashed line). Minimum and maximum errors in estimated taper curves are shown of corresponding color in dotted lines, except for existing model in solid lines.

**Figure 10.**Reference stem volume and predicted stem volume based on the taper curve equation that was parametrized using all (n = 246) Scots pines (

**a**), or random sample of 96 (

**b**) and 1 Scots pine (

**c**). Additionally, predicted volume includes random sampling of 100 Scots pine for validation to show the variation in the stem-volume estimates. Black line presents 1:1 relationship.

**Table 1.**Descriptive statistics of diameter-at-breast height (DBH) and height for all Scots pines within the 37 sample plots (n = 2351), the modelling (n = 246), and destructively (n = 19) pines. Min = minimum, 1Q = 25% percentile, 3Q = 75% percentile, max = maximum.

Attribute | Min | 1Q | Median | Mean | 3Q | Max | Standard Deviation | |
---|---|---|---|---|---|---|---|---|

All Scots pines | DBH (cm) | 4.8 | 15.7 | 19.9 | 19.4 | 23.5 | 39.8 | 6.2 |

Height (m) | 1.6 | 15.0 | 17.9 | 17.5 | 20.8 | 31.6 | 4.4 | |

Modelling pines | DBH (cm) | 5.0 | 18.9 | 22.0 | 22.3 | 25.8 | 39.8 | 5.6 |

Height (m) | 4.5 | 16.5 | 19.3 | 19.2 | 22.0 | 28.4 | 3.8 | |

Destructively measured pines | DBH (cm) | 10.3 | 17.1 | 19.4 | 19.6 | 23.0 | 25.9 | 4.0 |

Height (m) | 13.8 | 17.4 | 18.3 | 18.8 | 20.5 | 23.0 | 2.4 |

**Table 2.**Absolute and relative root mean square error (RMSE) and mean difference of volume estimates based on parametrized taper curves obtained with samples of varying sizes that were randomly selected 100 times. Min = minimum, Max = maximum, Mean = average, Std = standard deviation.

RMSE | Mean Difference | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

dm^{3} | % | dm^{3} | % | ||||||||||||||

min | max | mean | std | min | max | mean | std | min | max | mean | std | min | max | mean | std | ||

Sample Size | 1 | 29.2 | 156.3 | 71.6 | 30.0 | 12.0 | 62.5 | 29.6 | 12.3 | −126.3 | 70.5 | −31.6 | 39.3 | −49.3 | 28.6 | −13.1 | 16.3 |

5 | 27.0 | 109.7 | 57.4 | 18.7 | 11.8 | 46.7 | 23.6 | 7.2 | −76.7 | 21.3 | −26.0 | 20.2 | −31.0 | 9.7 | −10.7 | 8.2 | |

10 | 28.8 | 130.7 | 51.8 | 14.8 | 12.6 | 49.5 | 21.3 | 5.6 | −56.8 | 15.4 | −21.1 | 14.3 | −23.1 | 5.9 | −8.7 | 5.8 | |

15 | 28.4 | 119.3 | 51.4 | 14.4 | 12.3 | 45.2 | 21.2 | 5.5 | −48.7 | 3.0 | −21.9 | 10.9 | −19.4 | 1.2 | −9.0 | 4.4 | |

21 | 28.4 | 119.1 | 51.2 | 14.1 | 11.8 | 45.1 | 21.1 | 5.4 | −48.4 | 4.5 | −21.9 | 10.3 | −20.6 | 1.8 | −9.1 | 4.2 | |

46 | 29.3 | 117.8 | 50.5 | 13.5 | 12.1 | 44.6 | 20.8 | 5.0 | −45.6 | 0.7 | −21.6 | 8.4 | −17.8 | 0.3 | −8.9 | 3.2 | |

71 | 28.9 | 116.0 | 50.7 | 13.3 | 12.6 | 43.9 | 20.8 | 4.9 | −44.2 | −5.8 | −21.8 | 7.8 | −17.8 | −2.5 | −8.9 | 3.0 | |

96 | 32.6 | 117.2 | 50.8 | 13.1 | 13.5 | 44.4 | 20.9 | 4.8 | −45.6 | −9.2 | −22.2 | 7.1 | −17.3 | −4.0 | −9.1 | 2.7 | |

121 | 31.1 | 116.2 | 50.6 | 13.2 | 12.9 | 44.0 | 20.8 | 4.8 | −43.2 | −8.2 | −22.1 | 6.8 | −18.4 | −3.4 | −9.1 | 2.6 | |

146 | 30.7 | 115.3 | 50.6 | 13.4 | 12.7 | 43.7 | 20.8 | 4.9 | −46.4 | −7.7 | −22.0 | 7.3 | −19.1 | −3.2 | −9.1 | 2.8 | |

171 | 29.8 | 117.9 | 50.4 | 13.4 | 12.4 | 44.6 | 20.7 | 4.9 | −46.1 | −5.0 | −21.7 | 6.9 | −17.3 | −2.1 | −8.9 | 2.6 | |

196 | 30.9 | 121.0 | 50.4 | 13.4 | 12.8 | 45.8 | 20.7 | 4.9 | −48.9 | −8.3 | −21.8 | 6.8 | −18.5 | −3.4 | −8.9 | 2.6 | |

221 | 31.7 | 119.5 | 50.4 | 13.3 | 13.1 | 45.3 | 20.7 | 4.9 | −46.8 | −9.5 | −21.8 | 6.5 | −17.7 | −3.9 | −9.0 | 2.4 | |

246 | 30.6 | 122.3 | 50.5 | 13.2 | 12.7 | 46.3 | 20.8 | 4.8 | −50.9 | −7.2 | −21.8 | 6.4 | −19.3 | −3.0 | −9.0 | 2.4 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Saarinen, N.; Kankare, V.; Pyörälä, J.; Yrttimaa, T.; Liang, X.; Wulder, M.A.; Holopainen, M.; Hyyppä, J.; Vastaranta, M.
Assessing the Effects of Sample Size on Parametrizing a Taper Curve Equation and the Resultant Stem-Volume Estimates. *Forests* **2019**, *10*, 848.
https://doi.org/10.3390/f10100848

**AMA Style**

Saarinen N, Kankare V, Pyörälä J, Yrttimaa T, Liang X, Wulder MA, Holopainen M, Hyyppä J, Vastaranta M.
Assessing the Effects of Sample Size on Parametrizing a Taper Curve Equation and the Resultant Stem-Volume Estimates. *Forests*. 2019; 10(10):848.
https://doi.org/10.3390/f10100848

**Chicago/Turabian Style**

Saarinen, Ninni, Ville Kankare, Jiri Pyörälä, Tuomas Yrttimaa, Xinlian Liang, Michael A. Wulder, Markus Holopainen, Juha Hyyppä, and Mikko Vastaranta.
2019. "Assessing the Effects of Sample Size on Parametrizing a Taper Curve Equation and the Resultant Stem-Volume Estimates" *Forests* 10, no. 10: 848.
https://doi.org/10.3390/f10100848