# Branch Development of Five-Year-Old Betula alnoides Plantations in Response to Planting Density

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Site and Design

#### 2.2. Measurements

#### 2.3. Data Analysis

_{bpt}= μ + ρ

_{D}+ λ

_{r}+ λ

_{rp}+ λ

_{rpt}(tree level)

_{bptb}= μ + ρ

_{D}+ λ

_{r}+ λ

_{rp}+ λ

_{rpt}+ λ

_{rptb}(branch level)

_{D}is the effect of planting density; λ

_{r}, λ

_{rp}, λ

_{rpt}, and λ

_{rptb}are the random effects for block (r), plot (p), tree (t), and branch (b), respectively. Restricted maximum likelihood estimation (REML) was used in the mixed-model analysis, and multiple range tests were performed between treatments with least significant difference (LSD). Since the branch portion was expressed as a proportion, a transformation using the arcsine square root function was performed before analysis.

^{2}was also calculated as the squared correlation between observations and predictions for GLM, and marginal R

^{2}(${R}_{m}^{2}$, only the fixed effects) as well as conditional R

^{2}(${R}_{c}^{2}$, both the fixed and random effects) for LMM. Simulation plots based on the fitting models were also presented so as to expressing the relationships between the dependent and all independent variables. All data analyses and modelling were performed using SPSS 21.0 for Windows (IBM-SPSS Inc., Chicago, IL, USA).

## 3. Results

#### 3.1. Tree Growth Performance

#### 3.2. Branch Quantity (Number, Proportion, Density)

#### 3.3. Branch Morphology (Diameter, Length, Angle, the Largest Diameter)

#### 3.4. Modeling and Simulation on Branch Attributes

#### 3.4.1. Branch Number

_{0}+ a

_{1}× ln(PD)

_{0}+ b

_{1}× ln(PD)

^{2}= 0.368 for NLB, R

^{2}= 0.308 for NDB). The model precision was not high, either (RMSE = 7.9 for NBL, RMSE = 4.1 for NDB).

#### 3.4.2. Branch Diameter

_{0}+ c

_{1}× BH + c

_{2}× BL + c

_{3}× BA + α

_{rp}+ α

_{rptb}

#### 3.4.3. Branch Length

_{0}+ d

_{1}× BH + d

_{2}× BA + d

_{3}× BD + β

_{rp}+ β

_{rptb}

#### 3.4.4. Branch Angle

_{0}+ e

_{1}× BH + e

_{2}× BD + γ

_{rp}+ γ

_{rptb}

## 4. Discussion

#### 4.1. Tree Growth Performance

#### 4.2. Branch Quantity

#### 4.3. Branch Morphology

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Predicted branch number (solid line) based on simulated equations of live branch (Equation (3); (

**A**)) and dead branch (Equation (4); (

**B**)) with 95% confidence intervals (dashed lines), the solid black spots were the observed data of branch number.

**Figure 2.**Simulated plots (solid lines) based on marginal model (Equation (5)) for predicting branch diameter with 95% confidence intervals (dashed lines). (

**A**) Predicted branch diameter versus branch height based on mean branch length and angle; (

**B**) predicted branch diameter versus branch length based on mean branch height and angle; and (

**C**) predicted branch diameter versus branch angle based on mean branch height and length.

**Figure 3.**Simulated plots (solid lines) based on marginal model (Equation (6)) for predicting branch length with 95% confidence intervals (dashed lines). (

**A**) Predicted branch length versus branch height based on mean branch diameter and angle; (

**B**) predicted branch length versus branch diameter based on mean branch height and angle; and (

**C**) predicted branch length versus branch angle based on mean branch height and diameter.

**Figure 4.**Simulated plots (solid lines) based on marginal model (Equation (7)) for predicting branch angle with 95% confidence intervals (dashed lines). (

**A**) Predicted branch angle versus branch height based on mean branch diameter; and (

**B**) predicted branch angle versus branch diameter based on mean branch angle.

Abbreviation | Attributes Represent | Precision |
---|---|---|

Stand attributes | ||

PD | Planting density treatments | / |

Tree attributes | ||

H | Tree height | 0.1 m |

DBH | Stem diameter at breast height | 0.1 cm |

Hcb | Height to crown base | 0.1 m |

CW | Crown diameter | 0.1 m |

NLB | Number of live branches | / |

NDB | Number of dead branches | / |

Branch attributes | ||

BD | Branch diameter | 0.01 mm |

BL | Branch length | 0.1 m |

BA | Branch angle | 1° |

BH | Branch height | 0.01 m |

Model descriptors | ||

a, b, c, d, e | Coefficients of fixed effects and intercepts | / |

r, rp, rpt, rptb | Subscripts for replicate, plot, tree and branch | / |

α, β, γ, δ | Variance components (random effects) | / |

Φ | Dispersion parameter | / |

ln() | Natural log-link | / |

RMSE | Root mean squared error | / |

R^{2}_{(c)}, R^{2}_{(m)} | Conditional and marginal R^{2} | / |

**Table 2.**Plot and sampled dominant and co-dominant tree attributes of Betula alnoides. For abbreviations see Table 1.

Planting Density (sph) | Spacing (Row × Tree) | DBH (cm) | H (m) | Hcb (m) | CD (m) | |
---|---|---|---|---|---|---|

Stand level | 625 | 4 m × 4 m | 6.85 (0.21)a | 5.16 (0.15)b | 1.46 (0.06)c | 3.33 (0.08)a |

833 | 4 m × 3 m | 6.51 (0.13)a | 5.00 (0.09)b | 1.40 (0.04)c | 3.07 (0.06)b | |

1111 | 3 m × 3 m | 6.70 (0.14)a | 5.33 (0.11)b | 1.59 (0.04)bc | 2.81 (0.05)c | |

1250 | 4 m × 2 m | 6.65 (0.14)a | 5.65 (0.10)a | 1.80 (0.06)ab | 2.64 (0.05)d | |

1667 | 3 m × 2 m | 6.93 (0.14)a | 5.69 (0.12)a | 1.94 (0.08)a | 2.63 (0.04)d | |

2500 | 2 m × 2 m | 6.46 (0.13)a | 5.21 (0.10)b | 1.89 (0.11)a | 2.43 (0.05)e | |

Sampled dominant and co-dominant trees | 625 | 4 m × 4 m | 8.19 (0.48)a | 7.09 (0.26)a | 1.48 (0.09)b | 3.82 (0.15)a |

833 | 4 m × 3 m | 7.70 (0.21)a | 7.00 (0.28)a | 1.66 (0.25)ab | 3.56 (0.09)ab | |

1111 | 3 m × 3 m | 7.84 (0.28)a | 6.41 (0.16)a | 1.67 (0.10)ab | 3.42 (0.16)ab | |

1250 | 4 m × 2 m | 7.73 (0.24)a | 7.46 (0.20)a | 2.09 (0.09)a | 3.55 (0.09)ab | |

1667 | 3 m × 2 m | 7.79 (0.28)a | 7.46 (0.17)a | 1.97 (0.15)a | 3.34 (0.10)bc | |

2500 | 2 m × 2 m | 8.13 (0.41)a | 7.62 (0.16)a | 2.52 (0.19)a | 3.11 (0.09)c |

**Table 3.**Summary statistics of main branch attributes for Betula alnoides under six treatments of planting density.

Branch Attributes | Planting Density (sph) | ||||||
---|---|---|---|---|---|---|---|

625 | 833 | 1111 | 1250 | 1667 | 2500 | ||

Branch number | Live | 63 (3)a | 64 (3)a | 57 (3)ab | 51 (1)bc | 50 (3)bc | 47 (2)c |

Dead | 5 (1)d | 6 (1)cd | 6 (1)cd | 10 (2)b | 8 (1)bc | 13 (2)a | |

Total | 68 (3)a | 70 (3)a | 64 (4)a | 61 (2)a | 58 (2)a | 61 (2)a | |

Branch proportion | Live | 0.93 (0.01)a | 0.92 (0.02)a | 0.90 (0.01)ab | 0.84 (0.02)c | 0.86 (0.03)bc | 0.78 (0.03)d |

Dead | 0.07 (0.01)d | 0.08 (0.02)d | 0.10 (0.01)cd | 0.16 (0.02)ab | 0.14 (0.03)bc | 0.22 (0.03)a | |

Branch density | Live | 11.4 (0.9)a | 12.1 (0.8)a | 12.1 (0.6)a | 9.6 (0.5)b | 9.1 (0.5)b | 9.4 (0.5)b |

Dead | 0.8 (0.1)d | 1.0 (0.2)cd | 1.3 (0.1)cd | 1.8 (0.3)b | 1.5 (0.3)bc | 2.6 (0.3)a | |

Total | 12.3 (0.9)a | 13.2 (0.7)a | 13.4 (0.6)a | 11.4 (0.4)b | 10.6 (0.3)b | 11.9 (0.4)ab | |

Branch diameter (mm) | 12.11 (0.21)a | 11.21 (0.19)a | 11.04 (0.21)a | 11.85 (0.24)a | 11.05 (0.22)a | 11.15 (0.25)a | |

Branch angle (°) | 71.9 (0.6)a | 70.3 (0.7)a | 69.3 (0.7)a | 69.5 (0.8)a | 69.2 (1.0)a | 69.0 (0.9)a | |

Branch length (m) | 1.31 (0.02)a | 1.23 (0.02)a | 1.21 (0.02)a | 1.26 (0.02)a | 1.23 (0.03)a | 1.18 (0.03)a | |

The largest branch dimeter (mm) | 21.75 (0.69)a | 19.94 (0.64)ab | 19.05 (1.05)b | 19.35 (0.47)b | 18.06 (0.51)b | 18.45 (0.86)b | |

Branch height of the largest branch (m) | 4.00 (0.39)b | 3.80 (0.26)b | 4.18 (0.06)ab | 4.91 (0.28)a | 4.72 (0.25)a | 4.90 (0.25)a |

**Table 4.**Summary statistics of branch attributes at four orientations for Betula alnoides under six treatments of planting density.

Branch Attributes | Planting Density (sph) | ||||||
---|---|---|---|---|---|---|---|

625 | 833 | 1111 | 1250 | 1667 | 2500 | ||

Branch number | East | 12 (1) | 11 (1) | 10 (1) | 10 (0) | 7 (1) | 7 (1) |

South | 11 (1) | 13 (1) | 11 (1) | 9 (1) | 9 (1) | 8 (1) | |

West | 12 (1) | 10 (1) | 10 (1) | 8 (1) | 7 (1) | 7 (1) | |

North | 11 (1) | 12 (0) | 11 (1) | 9 (1) | 8 (1) | 8 (1) | |

Branch proportion | East | 0.26 (0.01) | 0.23 (0.01) | 0.24 (0.01) | 0.29 (0.01) | 0.23 (0.03) | 0.24 (0.02) |

South | 0.23 (0.02) | 0.29 (0.02) | 0.27 (0.02) | 0.24 (0.02) | 0.29 (0.03) | 0.26 (0.02) | |

West | 0.26 (0.02) | 0.22 (0.01) | 0.24 (0.02) | 0.21 (0.02) | 0.21 (0.03) | 0.23 (0.02) | |

North | 0.24 (0.02) | 0.26 (0.01) | 0.25 (0.02) | 0.26 (0.02) | 0.26 (0.02) | 0.27(0.03) |

**Table 5.**Parameter estimates of fixed and random variables for Equations (3)–(7). Abbreviations see Table 1.

Equation | Response Variables | Regression Parameters | Predictor Variables | Estimate | Standard Error | Significance |
---|---|---|---|---|---|---|

(3) | NLB | a_{0} | Intercept | 5.738 | 0.2973 | <0.0001 |

a_{1} | PD | −0.244 | 0.0421 | <0.0001 | ||

Φ | 1.125 | |||||

(4) | NDB | b_{0} | Intercept | −3.166 | 0.7910 | <0.0001 |

b_{1} | PD | 0.732 | 0.1089 | <0.0001 | ||

Φ | 2.051 | |||||

(5) | BD | c_{0} | Intercept | 1.4341 | 0.0331 | <0.0001 |

c_{1} | BH | 0.0489 | 0.0051 | <0.0001 | ||

c_{2} | BL | 0.7283 | 0.0128 | <0.0001 | ||

c_{3} | BA | −0.0023 | 0.0004 | <0.0001 | ||

α_{bp} | 0.0011 | 0.0006 | 0.041 | |||

α_{bptb} | 0.0440 | 0.0015 | <0.0001 | |||

(6) | BL | d_{0} | Intercept | −1.0484 | 0.0423 | <0.0001 |

d_{1} | BH | 0.0196 | 0.0058 | <0.0001 | ||

d_{2} | BA | 0.0025 | 0.0004 | <0.0001 | ||

d_{3} | BD | 0.0827 | 0.0016 | <0.0001 | ||

β_{bp} | 0.0017 | 0.0008 | 0.037 | |||

β_{bptb} | 0.0541 | 0.0018 | <0.0001 | |||

(7) | BA | e_{0} | Intercept | 4.0002 | 0.0331 | <0.0001 |

e_{1} | BH | 0.0726 | 0.0051 | <0.0001 | ||

e_{2} | BD | −0.0054 | 0.0014 | <0.0001 | ||

γ_{bp} | 0.0021 | 0.0010 | 0.027 | |||

γ_{bptb} | 0.0466 | 0.0015 | <0.0001 |

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## Share and Cite

**MDPI and ACS Style**

Wang, C.-S.; Tang, C.; Hein, S.; Guo, J.-J.; Zhao, Z.-G.; Zeng, J. Branch Development of Five-Year-Old *Betula alnoides* Plantations in Response to Planting Density. *Forests* **2018**, *9*, 42.
https://doi.org/10.3390/f9010042

**AMA Style**

Wang C-S, Tang C, Hein S, Guo J-J, Zhao Z-G, Zeng J. Branch Development of Five-Year-Old *Betula alnoides* Plantations in Response to Planting Density. *Forests*. 2018; 9(1):42.
https://doi.org/10.3390/f9010042

**Chicago/Turabian Style**

Wang, Chun-Sheng, Cheng Tang, Sebastian Hein, Jun-Jie Guo, Zhi-Gang Zhao, and Jie Zeng. 2018. "Branch Development of Five-Year-Old *Betula alnoides* Plantations in Response to Planting Density" *Forests* 9, no. 1: 42.
https://doi.org/10.3390/f9010042