# Occurrence of Density-Dependent Height Repression within Jack Pine and Black Spruce Populations

## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

#### 2.1. Detecting Density-Dependent Height Development

#### 2.1.1. Nelder Plot Structure and Measurements

_{d}; m). Note, the configuration of a Nelder plot does not allow for the explicit estimation of a population-based H

_{d}value nor enables the potential effects of plot size, spatial pattern and intraspecific effects on mean dominant height estimation to be explicitly accounted for (sensu [17,18]). Consequently, in this study, H

_{d}calculated in the fashion described was assumed to be conceptually similar to the traditionally caculated mean dominant height (e.g., mean dominant height of the trees within upper quintile of the height or diameter distribution) in respect to providing a measure of central tendency of the tallest tree for a given initial spacing treatment, measurement year, species and plot.

#### 2.1.2. Stand Density-Height Regression Relationships

_{e}is the nominal stand density equivalent at the time of establishment (stems/ha), and u is a composite error term assuming the presence of a 1st order autocorrelation scheme ($u={\widehat{p}}_{1}{u}_{a+1}+{v}_{a}$ where ${\widehat{p}}_{1}$ is the 1st order autocorrelation coefficient (−1 ≤ ${\widehat{p}}_{1}$ ≤ 1)) and v

_{a}is a random error term). The second step consisted of obtaining the residual at each concentric arc position (e

_{a}) and its corresponding adjacent one (e

_{a +}

_{1}) for each significant (p ≤ 0.05) relationship and subsequently fitting the ${e}_{a}={\widehat{p}}_{1}{e}_{a+1}+{v}_{a}$ regression model. The third step consisted of interpreting the frequency of significant ${e}_{a}={\widehat{p}}_{1}{e}_{a+1}+{v}_{a}$ relationships and assessing the magnitude of ${\widehat{p}}_{1}$, in order to determine if remedial parameterization methods were required.

## 3. Results

#### 3.1. Experimental Evidence of a Density-Dependent Height Repression Effect

**Figure 1.**Box and whisker plots graphically illustrating the observed relationship between initial spacing and mean height at approximately 40 year post-establishment by species and plot series. Note, the median value is denoted by the solid square within the open rectangle, first and third quartiles are denoted by the lower and upper horizontal sides of the open rectangle, respectively, and minimum and maximum values are denoted by the end points of the lower and upper whiskers, respectively.

#### 3.2. Quantifying the Density-Dependent Height Repression Effect

**Figure 2.**Box and whisker plots graphically illustrating the observed relationship between initial spacing and site productivity by species and plot series. Note, box and whisker plot denotations are given in Figure 1.

Species | Age | Nelder Plot Series | Spatial Autocorrelation Detection via Time Series Statistics ^{a} | Spatial Autocorrelation Detection via Regression Statistics ^{b} | ||
---|---|---|---|---|---|---|

Frequency of significant partial autocorrelation coefficients (n_{s}/n_{t}) | Spatial lag | Frequency of significant density-dependent height repression effects (n’_{s}/n_{t}) | Frequency of significant spatial autocorrelation (n’’_{s}/n’_{s}) | |||

Jack | 16 | Dunmore | 1/10 | 1st | 3/10 | 1/3 |

Pine | Willison | 3/9 | 1st | 6/9 | 0/6 | |

Terry | 1/9 | 1st | 5/9 | 0/5 | ||

20 | Dunmore | 1/10 | 1st | 3/10 | 1/3 | |

Willison | 2/9 | 1st | 7/9 | 0/7 | ||

Terry | 5/9 | 1st | 6/9 | 1/6 | ||

40 | Dunmore | 3/10 | 1st | 8/10 | 0/8 | |

Willison | 2/9 | 1st | 8/9 | 2/8 | ||

Terry | 4/8 | 1st | 8/8 | 1/8 | ||

Black | 41 | Dunmore | 3/10 | 1st | 8/10 | 0/8 |

Spruce | Willison | 3/9 | 1st | 8/9 | 0/8 | |

Terry | 3/8 | 1st | 8/8 | 0/8 |

^{a}Ratio of the number of significant (p ≤ 0.05) partial autocorrelation coefficients (n

_{s}) to the total number of plots assessed (n

_{t}) within a specified Nelder plots series at a specified measurement age;

^{b}(1) Ratio of the number of significant (p ≤ 0.05) mean dominant height—initial density regression relationships (n’

_{s}) to the total number of plots assessed (n

_{t}) within a specified Nelder plots series at a specified measurement age; and (2) Ratio of the number of significant (p ≤ 0.05) residual regressions (n’’

_{s}) to the total number of significant relationships (n’

_{s}) found in (1) for a given plot series and measurement age.

_{d}-N

_{e}relationships. The regression equations explained a moderate level of variation: mean (minimum/maximum) coefficient of determination (r

^{2}) values of 0.512 (0.279/0.760), 0.560 (0.312/0.854), 0.683 (0.387/0.889) for jack pine at ages 16, 20 and 40, respectively, and 0.602 (0.373/0.919) for black spruce at age 41. The magnitude of error was acceptable as reflected by moderate levels of the standard error of the estimate: mean (minimum/maximum) of 0.320 m (0.190/0.455), 0.439 m (0.218/0.736), 0.866 m (0.434/1.270) for jack pine at age 16, 20 and 40, respectively, and 0.834 m (0.369/1.399) for black spruce at age 41. Overall, the regression relationships were in general compliance with the constant variance and normality assumptions underlying OLS parameterization (e.g., horizontal band and invariant pattern of raw residuals when plotted against the predictor variable).

**Figure 3.**Bivariate scatter plots illustrating the relationship between initial density and mean height for (

**a**) jack pine at approximately 16, 20 and 40 years post-establishment for each Nelder plot series; (

**b**) black spruce at approximately 40 years post-establishment for each Nelder plot series; and (

**c**) jack pine and black spruce at approximately 40 years post-establishment for all the Nelder plots combined. The whiskers defined the mean ± standard deviation (SD) values and the dotted line is the estimated height-density relationship for the height repression coefficients given in Table 2.

**Table 2.**Resultant mean dominant height-density relationship by species, age and plot series inclusive of the density-dependent height repression coefficient estimates and their 95% confidence limits.

Species | Age | Plot Series | n | Intercept | Density-Dependent Height Repression Coefficient ^{a} | ||
---|---|---|---|---|---|---|---|

95% CI-L | ${\widehat{\beta}}_{1}^{*}$ | 95% CI-U | |||||

Jack Pine | 16 | Dunmore | 3 | 6.634 | −0.000130 | −0.000036 | 0.000058 |

Willison | 6 | 7.028 | −0.000080 | −0.000038 | 0.000004 | ||

Terry | 5 | 6.019 | −0.000059 | −0.000026 | 0.000007 | ||

Combined | 14 | 6.563 | −0.000050 | −0.000031 | −0.000012 | ||

20 | Dunmore | 3 | 9.215 | −0.000216 | −0.000060 | 0.000096 | |

Willison | 7 | 9.993 | −0.000094 | −0.000047 | 0.000000 | ||

Terry | 6 | 8.840 | −0.000077 | −0.000036 | 0.000005 | ||

Combined | 16 | 9.384 | −0.000067 | −0.000042 | −0.000018 | ||

40 | Dunmore | 8 | 17.007 | −0.000256 | −0.000139 | −0.000021 | |

Willison | 8 | 17.032 | −0.000232 | −0.000124 | −0.000017 | ||

Terry | 8 | 16.385 | −0.000286 | −0.000154 | −0.000023 | ||

Combined | 24 | 16.789 | −0.000200 | −0.000137 | −0.000078 | ||

Black Spruce | 41 | Dunmore | 8 | 11.300 | −0.000202 | −0.000108 | −0.000015 |

Willison | 8 | 12.367 | −0.000198 | −0.000106 | −0.000014 | ||

Terry | 8 | 10.624 | −0.000185 | −0.000099 | −0.000014 | ||

Combined | 24 | 11.429 | −0.000150 | −0.000104 | −0.000059 |

^{a}Grand mean density-dependent height repression coefficient as calculated by Equation (1) and associated lower and upper 95% confidence limits (denoted CI-L and CI-U, respectively).

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix

**Table A1.**Plot-level species-specific parameter estimates and associated regression statistics for significant (p ≤ 0.05) mean dominant height-density relationships by measurement age.

Species | Age | Nelder Plot Series | Plot No. | Parameter Estimates ^{a} | Regression Statistics ^{b} | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Intercept | Slope (height-repression coefficient) | DF | r^{2} | SEE (m) | F-ratio | |||||||

Estimate | SE | Estimate | 95% CL | n_{reg}, n_{res} | ||||||||

Lower | Upper | |||||||||||

Jack Pine | 16 | Dunmore | 1 | 7.20 | 0.143 | −0.0000450 | −0.0000696 | −0.0000205 | 1,14 | 0.525 | 0.371 | 15.5 * |

5 | 7.65 | 0.175 | −0.0000554 | −0.0000856 | −0.0000253 | 1,14 | 0.526 | 0.455 | 15.5 * | |||

9 | 5.25 | 0.150 | −0.0000281 | −0.0000539 | −0.0000022 | 1,14 | 0.279 | 0.390 | 5.4 * | |||

Willison | 11 | 7.83 | 0.115 | −0.0000612 | −0.0000809 | −0.0000415 | 1,14 | 0.760 | 0.298 | 44.2 * | ||

12 | 6.91 | 0.114 | −0.0000260 | −0.0000459 | −0.0000061 | 1,13 | 0.381 | 0.296 | 8.0 * | |||

13 | 7.32 | 0.134 | −0.0000378 | −0.0000607 | −0.0000149 | 1,13 | 0.495 | 0.342 | 12.8 * | |||

14 | 6.97 | 0.164 | −0.0000593 | −0.0000876 | −0.0000310 | 1,14 | 0.590 | 0.427 | 20.2 * | |||

16 | 6.60 | 0.131 | −0.0000433 | −0.0000659 | −0.0000207 | 1,14 | 0.547 | 0.341 | 16.9 * | |||

17 | 6.99 | 0.125 | −0.0000450 | −0.0000666 | −0.0000234 | 1,14 | 0.588 | 0.326 | 20.0 * | |||

Terry | 22 | 7.18 | 0.112 | −0.0000399 | −0.0000593 | −0.0000205 | 1,14 | 0.582 | 0.292 | 19.5 * | ||

23 | 5.88 | 0.108 | −0.0000265 | −0.0000451 | −0.0000079 | 1,14 | 0.401 | 0.280 | 9.4 * | |||

25 | 5.55 | 0.092 | −0.0000205 | −0.0000363 | −0.0000046 | 1,14 | 0.354 | 0.239 | 7.7 * | |||

26 | 5.56 | 0.073 | −0.0000314 | −0.0000440 | −0.0000188 | 1,14 | 0.671 | 0.190 | 28.5 * | |||

28 | 6.05 | 0.093 | −0.0000250 | −0.0000407 | −0.0000093 | 1,13 | 0.476 | 0.232 | 11.8 * | |||

20 | Dunmore | 1 | 9.64 | 0.209 | −0.0000604 | −0.0000965 | −0.0000243 | 1,14 | 0.479 | 0.544 | 12.9 * | |

5 | 10.53 | 0.283 | −0.0001142 | −0.0001629 | −0.0000654 | 1,14 | 0.643 | 0.736 | 25.2 * | |||

9 | 7.88 | 0.193 | −0.0000495 | −0.0000827 | −0.0000162 | 1,14 | 0.421 | 0.501 | 10.2 * | |||

Willison | 11 | 10.99 | 0.188 | −0.0000848 | −0.0001171 | −0.0000525 | 1,14 | 0.693 | 0.488 | 31.7 * | ||

12 | 9.87 | 0.129 | −0.0000287 | −0.0000512 | −0.0000062 | 1,13 | 0.369 | 0.335 | 7.6 * | |||

13 | 10.29 | 0.161 | −0.0000435 | −0.0000709 | −0.0000161 | 1,13 | 0.475 | 0.410 | 11.8 * | |||

14 | 10.11 | 0.181 | −0.0000930 | −0.0001241 | −0.0000619 | 1,14 | 0.746 | 0.469 | 41.1 * | |||

15 | 10.87 | 0.241 | −0.0000655 | −0.0001070 | −0.0000240 | 1,14 | 0.450 | 0.626 | 11.5 * | |||

16 | 9.06 | 0.152 | −0.0000611 | −0.0000873 | −0.0000349 | 1,14 | 0.640 | 0.396 | 24.9 * | |||

17 | 9.90 | 0.160 | −0.0000694 | −0.0000969 | −0.0000418 | 1,14 | 0.676 | 0.416 | 29.2 * | |||

Terry | 22 | 10.33 | 0.214 | −0.0000606 | −0.0000975 | −0.0000237 | 1,14 | 0.470 | 0.557 | 12.4 * | ||

23 | 8.78 | 0.125 | −0.0000367 | −0.0000582 | −0.0000151 | 1,14 | 0.487 | 0.325 | 13.3 * | |||

25 | 8.46 | 0.084 | −0.0000610 | −0.0000755 | −0.0000466 | 1,14 | 0.854 | 0.218 | 82.0 * | |||

26 | 8.52 | 0.126 | −0.0000579 | −0.0000796 | −0.0000362 | 1,14 | 0.700 | 0.327 | 32.7 * | |||

28 | 8.92 | 0.120 | −0.0000227 | −0.0000430 | −0.0000025 | 1,13 | 0.312 | 0.298 | 5.9 * | |||

29 | 8.72 | 0.146 | −0.0000481 | −0.0000732 | −0.0000230 | 1,14 | 0.547 | 0.378 | 16.9 * | |||

40 | Dunmore | 1 | 17.55 | 0.392 | −0.0001911 | −0.0002585 | −0.0001236 | 1,14 | 0.725 | 1.018 | 36.9 * | |

2 | 16.68 | 0.321 | −0.0001829 | −0.0002396 | −0.0001261 | 1,13 | 0.788 | 0.833 | 48.5 * | |||

3 | 17.16 | 0.337 | −0.0001396 | −0.0001992 | −0.0000799 | 1,13 | 0.663 | 0.875 | 25.6 * | |||

4 | 18.31 | 0.471 | −0.0001326 | −0.0002184 | −0.0000469 | 1,13 | 0.462 | 1.221 | 11.2 * | |||

6 | 17.63 | 0.288 | −0.0001510 | −0.0002006 | −0.0001013 | 1,14 | 0.753 | 0.748 | 42.6 * | |||

7 | 16.51 | 0.230 | −0.0001383 | −0.0001779 | −0.0000988 | 1,14 | 0.801 | 0.597 | 56.2 * | |||

8 | 16.26 | 0.406 | −0.0001199 | −0.0001897 | −0.0000500 | 1,14 | 0.492 | 1.054 | 13.5 * | |||

10 | 16.59 | 0.489 | −0.0001166 | −0.0002008 | −0.0000324 | 1,14 | 0.387 | 1.270 | 8.8 * | |||

Willison | 11 | 18.16 | 0.353 | −0.0001393 | −0.0002002 | −0.0000785 | 1,14 | 0.633 | 0.918 | 24.1 * | ||

12 | 17.41 | 0.288 | −0.0000896 | −0.0001397 | −0.0000395 | 1,13 | 0.535 | 0.746 | 14.9 * | |||

13 | 17.64 | 0.235 | −0.0001261 | −0.0001661 | −0.0000861 | 1,13 | 0.781 | 0.599 | 46.4 * | |||

14 | 16.84 | 0.381 | −0.0001350 | −0.0002007 | −0.0000694 | 1,14 | 0.582 | 0.990 | 19.5 * | |||

16 | 16.58 | 0.286 | −0.0001943 | −0.0002435 | −0.0001451 | 1,13 | 0.848 | 0.737 | 72.7 * | |||

17 | 16.84 | 0.293 | −0.0001993 | −0.0002551 | −0.0001434 | 1,13 | 0.821 | 0.749 | 59.4 * | |||

18 | 16.63 | 0.333 | −0.0001078 | −0.0001657 | −0.0000499 | 1,13 | 0.554 | 0.863 | 16.1 * | |||

19 | 17.25 | 0.376 | −0.0001349 | −0.0001996 | −0.0000702 | 1,13 | 0.610 | 0.968 | 20.3 * | |||

Terry | 23 | 17.11 | 0.448 | −0.0001933 | −0.0002705 | −0.0001160 | 1,14 | 0.673 | 1.165 | 28.8 * | ||

24 | 15.94 | 0.438 | −0.0001668 | −0.0002422 | −0.0000914 | 1,14 | 0.616 | 1.138 | 22.5 * | |||

25 | 16.48 | 0.267 | −0.0002068 | −0.0002527 | −0.0001608 | 1,14 | 0.869 | 0.693 | 93.1 * | |||

26 | 16.46 | 0.229 | −0.0001492 | −0.0001908 | −0.0001075 | 1,13 | 0.821 | 0.593 | 59.8 * | |||

27 | 16.54 | 0.168 | −0.0001339 | −0.0001630 | −0.0001047 | 1,13 | 0.883 | 0.434 | 98.4 * | |||

28 | 17.10 | 0.264 | −0.0002240 | −0.0002694 | −0.0001785 | 1,14 | 0.889 | 0.685 | 111.8 * | |||

29 | 16.19 | 0.388 | −0.0001184 | −0.0001872 | −0.0000496 | 1,13 | 0.515 | 1.009 | 13.8 * | |||

30 | 16.36 | 0.347 | −0.0001500 | −0.0002092 | −0.0000909 | 1,13 | 0.698 | 0.886 | 30.0 * | |||

Black Spruce | 41 | Dunmore | 1 | 11.33 | 0.210 | −0.0001484 | −0.0001867 | −0.0001101 | 1,13 | 0.844 | 0.545 | 70.2 * |

2 | 10.84 | 0.142 | −0.0000723 | −0.0000968 | −0.0000479 | 1,14 | 0.742 | 0.369 | 40.2 * | |||

4 | 13.04 | 0.433 | −0.0001076 | −0.0001821 | −0.0000331 | 1,14 | 0.407 | 1.124 | 9.6 * | |||

6 | 12.11 | 0.311 | −0.0001176 | −0.0001712 | −0.0000640 | 1,14 | 0.613 | 0.809 | 22.1 * | |||

7 | 11.13 | 0.338 | −0.0001390 | −0.0001972 | −0.0000808 | 1,14 | 0.652 | 0.878 | 26.2 * | |||

8 | 10.90 | 0.256 | −0.0001135 | −0.0001576 | −0.0000695 | 1,14 | 0.686 | 0.665 | 30.5 * | |||

9 | 10.95 | 0.370 | −0.0001124 | −0.0001761 | −0.0000487 | 1,14 | 0.506 | 0.961 | 14.3 * | |||

10 | 11.26 | 0.253 | −0.0001806 | −0.0002243 | −0.0001370 | 1,14 | 0.849 | 0.658 | 78.9 * | |||

Willison | 11 | 13.09 | 0.435 | −0.0001188 | −0.0001938 | −0.0000438 | 1,14 | 0.452 | 1.131 | 11.6 * | ||

12 | 12.67 | 0.375 | −0.0001805 | −0.0002451 | −0.0001159 | 1,14 | 0.720 | 0.975 | 35.9 * | |||

14 | 11.33 | 0.465 | −0.0001129 | −0.0001930 | −0.0000328 | 1,14 | 0.395 | 1.209 | 9.1 * | |||

15 | 13.04 | 0.178 | −0.0000712 | −0.0001019 | −0.0000405 | 1,14 | 0.639 | 0.463 | 24.7 * | |||

16 | 12.95 | 0.179 | −0.0001717 | −0.0002023 | −0.0001412 | 1,13 | 0.919 | 0.457 | 147.8 * | |||

17 | 11.37 | 0.407 | −0.0000943 | −0.0001644 | −0.0000243 | 1,14 | 0.373 | 1.057 | 8.3 * | |||

18 | 12.97 | 0.290 | −0.0001332 | −0.0001831 | −0.0000833 | 1,14 | 0.701 | 0.753 | 32.8 * | |||

19 | 12.92 | 0.473 | −0.0001140 | −0.0001954 | −0.0000325 | 1,14 | 0.391 | 1.229 | 9.0 * | |||

Terry | 23 | 10.61 | 0.290 | −0.0000926 | −0.0001425 | −0.0000427 | 1,14 | 0.531 | 0.752 | 15.9 * | ||

24 | 10.78 | 0.293 | −0.0001468 | −0.0001973 | −0.0000963 | 1,14 | 0.735 | 0.762 | 38.8 * | |||

25 | 10.23 | 0.538 | −0.0001838 | −0.0002765 | −0.0000911 | 1,14 | 0.564 | 1.399 | 18.1 * | |||

26 | 10.72 | 0.222 | −0.0001025 | −0.0001402 | −0.0000648 | 1,13 | 0.726 | 0.563 | 34.5 * | |||

27 | 10.19 | 0.261 | −0.0001130 | −0.0001578 | −0.0000681 | 1,14 | 0.676 | 0.677 | 29.2 * | |||

28 | 11.05 | 0.303 | −0.0000735 | −0.0001258 | −0.0000213 | 1,14 | 0.394 | 0.789 | 9.1 * | |||

29 | 11.33 | 0.355 | −0.0000989 | −0.0001601 | −0.0000378 | 1,14 | 0.462 | 0.923 | 12.0 * | |||

30 | 11.17 | 0.339 | −0.0000947 | −0.0001532 | −0.0000363 | 1,14 | 0.463 | 0.882 | 12.1 * |

^{a}OLS parameter estimates for the intercept $({\beta}_{0})$ with associated standard error (SE), and slope $({\beta}_{1})$ with associated lower and upper 95% confidence limits (95% CL).

^{b}Degrees of freedom (DF) for regression and residual error (n

_{reg}and n

_{res}, respectively), coefficient of determination (r

^{2}), standard error of the estimate (SEE), and F-statistic (F-ratio). Note, F-ratios superscripted by * denotes a significant (p ≤ 0.05) relationship.

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## Share and Cite

**MDPI and ACS Style**

Newton, P.F.
Occurrence of Density-Dependent Height Repression within Jack Pine and Black Spruce Populations. *Forests* **2015**, *6*, 2450-2468.
https://doi.org/10.3390/f6072450

**AMA Style**

Newton PF.
Occurrence of Density-Dependent Height Repression within Jack Pine and Black Spruce Populations. *Forests*. 2015; 6(7):2450-2468.
https://doi.org/10.3390/f6072450

**Chicago/Turabian Style**

Newton, Peter F.
2015. "Occurrence of Density-Dependent Height Repression within Jack Pine and Black Spruce Populations" *Forests* 6, no. 7: 2450-2468.
https://doi.org/10.3390/f6072450