# Climate Sensitive Tree Growth Functions and the Role of Transformations

## Abstract

**:**

_{2}concentration or changed management in terms of reduced nutrient subtractions from forest ground, since industrialization lowered the demand of residue and slash uptake.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Forest Research Data

^{2}and the standard interval length for re-measurement was 5 years. Immediately before inventories, managers mark trees for harvesting, so that the intensity and timing of the intervention is precisely recorded. The entire data set included 374 plots, at altitudes between 385 and 1930 m a.s.l., covering a wide range of climate types. They also cover the Swiss ecophysiological regions of Jura, Plateau, Pre-Alps and Alps. Their locations are spread all over Switzerland and are shown in Figure 1.

#### 2.2. Explanatory Variables

#### 2.2.1. Competition

#### 2.2.2. Stand Development

#### 2.2.3. Site

#### 2.2.4. Stand Density

#### 2.2.5. Thinning

#### 2.2.6. Mixture

#### 2.2.7. Climate

#### 2.2.8. Years Past 1940

#### 2.3. Model Fitting

#### 2.3.1. Base Model

#### 2.3.2. Transformation

- ${y}^{\prime}$ = sign(y) ${|y|}^{1/n}$, with $n=2,3,4$
- ${y}^{\prime}=log(y+\alpha )$, with $\alpha >min(y)$

#### 2.3.3. Model Selection

#### 2.3.4. Model Validation

## 3. Results

#### 3.1. Role of Transformation

#### 3.2. Climate Sensitive Growth Functions

#### 3.2.1. Overview

#### 3.2.2. Effects

#### Competition

#### Stand Development

^{2}was positive. For small diameters, the estimated growth is large for small ages, but heavily decreases with increasing age (the shown gradient is steep). The same trend is found with large diameters, but with a much smaller gradient. In the case of fir, $ddom$ was superior compared to age-related effects. This is reasonable, since fir is highly shade tolerant and thus age is potentially a bad measure of growth vigor. For small diameters, an increase in $ddom$ means slightly faster expected growth, while in larger trees this effect is stronger.

#### Site

#### Stand Density

#### Thinning

#### Mixture

#### Climate

#### Non Significant Effects

## 4. Discussion

#### 4.1. Transformation

#### 4.2. Estimation Method

#### 4.3. Estimated Effects

#### 4.3.1. Explained Variance

#### 4.3.2. Model Selection

#### 4.3.3. Nitrogen

#### 4.3.4. Years Past 1940

_{2}in the atmosphere, which can stimulate growth. Alternatively, it could be the result of changes in management. After industrialization, the pressure on forests to collect residues and slash clearly lessened, resulting in soils regenerating from the extraction of nutrients.

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

## Abbreviations

d | (single tree-specfic) tree diameter (at 1.3 m height, breast height) in cm |

$ddom$ | dominant diameter in cm: quadratic mean of the 100 thickest trees per hectare |

A | total plot area in hectares (10^{4} m^{2}) |

a | stand age in years |

$ba$ | tree basal area (at breast height) in m^{2}: $\frac{\pi}{40000}{d}^{2}$ |

$bat$ | total plot basal area in m^{2}/ha: $\frac{\pi}{4A}{\sum}_{i=1}^{n}{d}_{i}^{2}$ |

${t}_{i}$ | ith year of the measurement |

$bai$ | basal area increment normalized to 5-year period: $5\frac{\frac{\pi}{4}({d}_{{t}_{i}}^{2}-{d}_{{t}_{i-1}}^{2})}{{t}_{i}-{t}_{i-1}}$ |

$rbai$ | relative basal area increment: $bai/ba$ |

$cba$ | cumulative basal area: $cb{a}_{j}=\frac{1}{bat}{\sum}_{i=1}^{n}({\mathbf{1}}_{{d}_{i}\le {d}_{j}}b{a}_{i})$ |

$bal$ | basal area of larger trees: $ba{l}_{j}=\frac{1}{A}{\sum}_{i=1}^{n}({\mathbf{1}}_{{d}_{i}>{d}_{j}}b{a}_{i})$ |

$rbal$ | relative basal area of larger trees: $bal/bat$ |

$bah$ | percentage of basal area previously harvested in relation to $bat$ |

$nt$ | total number of trees per hectare in a plot |

$nh$ | percentage of stems harvested since last inventory |

$cs$ | crown surface area in m^{2}: $\pi \frac{\widehat{r}}{6{\widehat{l}}^{2}}\left({(4{\widehat{l}}^{2}+{\widehat{r}}^{2})}^{3/2}-{\widehat{r}}^{3}\right)$ with $\widehat{r}$ the imputed crown radius, $\widehat{l}$ the imputed crown length |

$cb$ | crown base (height where the living crown starts), in m |

$csa$ | crown cross sectional area in m^{2} |

$sdi$ | stand density index: $nt{(dq/25)}^{1.605}$ |

$cc$ | crown competition index Equation (2) |

$ccc$ | change in crown competition index after harvesting has taken place Equation (2) |

$lead.con$ | leading tree species is coniferous (holding maximum basal area in a stand) |

$year.past1940$ | if the observation year is before 1940, it is zero, otherwise the current year of observation−1940. |

p | yearly mean precipitation sum over period length in mm. Calculated for physiological year, meaning that the sum was taken from October until September, in contrast to calender years [42]. |

t | mean physiological year temperature, over intervals such as p. |

m | Aridity index (defined by deMartonne ([38], [p. 520])). $A=\frac{nphy.\ast}{tphy.\ast +10}$. Since this index means smaller values have higher aridity, its inverse interpretation is more intuitive. Consequently it was labeled “moisture-index” (m). In contrast to precipitation and temperature, moisture shows higher explanatory power in the spring months [42]; hence only the spring months (March–June) over periods were used to calculate the moisture index. |

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**Figure 1.**Locations of the Experimental Forest Management (EFM) trials in Switzerland, used in this study. Reproduced by permission of swisstopo (JA100118).

**Figure 2.**Observed basal area increment ($bai$) over diameter at breast height (d) for all species. Fifty-three percent of the variance is explained by a simple model (Equation (3), with $g=\sqrt{.}$ and f = Identity), using only d as explanatory variable. The blue background is the two-dimensional density of the observed d and $bai$. Dark blue contains more data, the 100 points indicate areas with the lowest density.

**Figure 3.**Residual sum of squares (in millions) for spruce and beech final models under different transformations of growth. Back-transformation was applied directly and with a correction factor [47]. The log-transformation depends on $\alpha $. Horizontal lines show the root-transformations, independent of $\alpha $. The fourth root transformations for the beech model are outside the plotting range. Their values are 88.5 (direct) and 89.8 (Snowdon-correction).

**Figure 5.**Effects of the stand development measured by age or ddom (quadratic mean diameter of dominant trees) on mean expected growth. The blue background is the two-dimensional density of the observed age and diameter values. Dark blue contains more data.

**Figure 6.**Effects of nitrogen deposition on basal area increment, per tree species. Only diameter and nitrogen deposition effects are shown, all other explanatory variables are held constant (mean).

**Figure 7.**Interacting effects of basal area total ($bat$) and total stem number per hectare ($nt$). The blue in the background is the two-dimensional density of the observed $bat$ and $nt$. Dark blue contains more data.

**Figure 8.**Predicted basal area increment for interacting effects of temperature (t) and precipitation (p). t and p are mean values over physiological years (see Abbreviations). The blue in the background is the two-dimensional density of the observed t and p. Dark blue contains more data.

**Figure 9.**Predicted basal area increment for interacting effects of temperature (t) and moisture (m). Temperature is the mean value over physiological years, moisture is the mean over spring months (see Abbreviations).

**Table 1.**Overview and quantiles of the data. The following abbreviations were used: Nor. = northness; Eas. = eastness; Y.P. 1940 = years past 1940.

Variable Units | bai m^{2}/(5 years) | d cm | bal m^{2} | rbal - | cba - | cc - | a years | ddom cm | csa m^{2} | ndep kg/ha/year | slope % | Nor. - | Eas. - | sdi - | bat m^{2} | nt 1/ha | bah % | nh % | ccc - | mixB - | t °C | p mm | m mm/°C | Y.P. 1940 years | year - |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

min | −29.8 | 0.6 | 0.0 | 0.0 | 0.0 | 0.0 | 22 | 11.4 | 3.1 | 5.2 | 2.5 | −1.0 | −1.0 | 57 | 3.5 | 24 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2 | 145 | 10.8 | 0 | 1904 |

5% | 0.2 | 6.2 | 4.3 | 0.1 | 0.0 | 0.5 | 34 | 16.4 | 23.6 | 11.1 | 2.5 | −1.0 | −1.0 | 489 | 17.9 | 342 | 0.0 | 0.0 | 0.0 | 0.0 | 3.4 | 747 | 14.4 | 0 | 1913 |

50% | 32.4 | 17.6 | 24.6 | 0.8 | 0.2 | 1.1 | 63 | 33.3 | 73.2 | 19.9 | 10.0 | 0.0 | 0.0 | 767 | 35.2 | 1050 | 0.1 | 0.1 | 0.1 | 0.1 | 7.9 | 1005 | 22.3 | 0 | 1936 |

95% | 181.0 | 46.4 | 46.9 | 1.0 | 0.6 | 2.4 | 154 | 58.2 | 294.0 | 37.1 | 50.0 | 1.0 | 0.7 | 1181 | 58.5 | 4060 | 0.2 | 0.3 | 0.3 | 1.0 | 9.1 | 1412 | 39.6 | 53 | 1993 |

max | 901.9 | 156.5 | 93.2 | 1.0 | 1.0 | 4.1 | 293 | 81.7 | 1206.9 | 60.6 | 80.0 | 1.0 | 1.0 | 1660 | 94.2 | 11,139 | 0.6 | 0.7 | 1.7 | 1.0 | 11.5 | 1693 | 54.0 | 72 | 2012 |

**Table 2.**Selected models and estimated coefficients (“Co.”). A blank space means that the variable was not part of the final model. The standard errors of the coefficients can be found in the Supplement “Supp1.docx”.

Group | Co. | Variable | Maple | Beech | Douglas Fir | Spruce | Pine | Larch | Oak | Fir |
---|---|---|---|---|---|---|---|---|---|---|

Diameter | ${\beta}_{0}$ | Intercept | 5.858 | 1.150 | 2.680 | 5.429 | −6.390 | 7.339 | −7.538 | −1.750 |

${\beta}_{1}$ | d | 16.13 | 20.30 | 36.18 | 19.08 | 19.89 | 21.12 | 21.66 | 19.48 | |

${\beta}_{2}$ | d.exp | −0.01931 | −0.03512 | −0.02289 | −0.03172 | −0.02540 | −0.02407 | −0.03159 | −0.02711 | |

${\beta}_{3}$ | bal | −0.060920 | −0.065076 | −0.075616 | ||||||

Competition | ${\beta}_{4}$ | rbal | −4.482 | −2.030 | ||||||

${\beta}_{5}$ | cba | 1.404 | 0.845 | 3.520 | ||||||

Stand development | ${\beta}_{7}$ | a | −0.05312 | −0.05535 | −0.34983 | −0.05592 | −0.08499 | −0.07945 | −0.06539 | |

${\beta}_{8}$ | a^{2} | 0.0002020 | 0.0002173 | 0.0015429 | 0.0001668 | 0.0002086 | 0.0001778 | 0.0001548 | ||

${\beta}_{9}$ | ddom | 0.07733 | ||||||||

${\beta}_{11}$ | ndep | 0.073038 | −0.127648 | −0.010480 | −0.188290 | −0.111304 | 0.011745 | −0.168438 | −0.158790 | |

${\beta}_{12}$ | ndep^{2} | 0.002485 | 0.002982 | 0.001768 | 0.000970 | 0.003283 | 0.002066 | |||

${\beta}_{19}$ | slope | −0.2442 | 0.0029 | |||||||

${\beta}_{20}$ | northness | −0.2485 | ||||||||

Site | ${\beta}_{21}$ | eastness | 1.3340 | 0.1786 | 0.4530 | |||||

${\beta}_{22}$ | relief.depres. | −0.9685 | ||||||||

${\beta}_{23}$ | relief.slope | −0.6542 | −0.4546 | 0.2175 | ||||||

${\beta}_{24}$ | relief.summit | −0.1927 | −0.8402 | 0.9349 | ||||||

${\beta}_{25}$ | relief.slope.toe | 0.6588 | −1.7911 | |||||||

Stand density | ${\beta}_{27}$ | sdi | −0.002934 | −0.003622 | ||||||

${\beta}_{28}$ | bat | −0.20652 | −0.07191 | −0.06243 | −0.05675 | 0.02064 | −0.02525 | |||

${\beta}_{29}$ | bat^{2} | 0.001587 | 0.000363 | −0.001909 | −0.000830 | |||||

${\beta}_{30}$ | bat$\xb7log10$(nt) | 0.00349 | 0.01100 | |||||||

${\beta}_{31}$ | $log10$(nt) | 0.6270 | 2.2025 | |||||||

Thinning | ${\beta}_{32}$ | treat | 0.5474 | 0.1521 | 1.7647 | 0.5355 | −0.2515 | |||

${\beta}_{33}$ | bah | 0.0208 | 1.9846 | |||||||

${\beta}_{35}$ | ccc | −2.209500 | ||||||||

Mixture | ${\beta}_{36}$ | mixB | −2.893 | −2.830 | −2.419 | −1.911 | ||||

${\beta}_{37}$ | lead.con | 1.405297 | 1.189001 | |||||||

Climate | ${\beta}_{38}$ | t | −0.3261 | 0.3434 | 1.9871 | 0.0866 | 2.4940 | −0.2539 | 3.0467 | −1.1924 |

${\beta}_{39}$ | p | 0.00170 | 0.00089 | 0.00070 | −0.00111 | −0.00272 | −0.00619 | 0.00116 | −0.00188 | |

${\beta}_{40}$ | m | 0.3422 | 0.1934 | −0.3176 | 0.1019 | 0.6144 | −0.2154 | −0.4253 | −0.1410 | |

${\beta}_{41}$ | $p\xb7m$ | −0.0000504 | −0.0001068 | −0.0011325 | 0.0008349 | −0.0000792 | ||||

${\beta}_{42}$ | $t\xb7m$ | 0.0545 | −0.0401 | 0.0363 | 0.0300 | |||||

${\beta}_{43}$ | t^{2} | −0.1672 | −0.0078 | −0.1312 | −0.0397 | −0.1637 | 0.0656 | |||

${\beta}_{44}$ | p^{2} | 0.00000205 | 0.00001383 | 0.00000274 | −0.00000949 | 0.00000163 | ||||

${\beta}_{45}$ | m^{2} | −0.00761932 | −0.00196531 | 0.00007983 | 0.01552329 | 0.00251696 | −0.00806958 | 0.00146262 | ||

Unexplained | ${\beta}_{46}$ | year.past1940 | 0.0298 | 0.0227 | 0.0924 | 0.0320 | 0.0295 | 0.0401 | 0.0196 |

**Table 3.**Summary statistics of the tree-specific models (back-transformed). $rmse(\%)=rmse/\overline{y}\xb7100\%.$

Tree species | Maple | Beech | Douglas | Spruce | Pine | Larch | Oak | Fir |
---|---|---|---|---|---|---|---|---|

n | 2450 | 149,009 | 9759 | 190,047 | 13,452 | 24,807 | 44,799 | 139,977 |

${R}_{in-sample}^{2}$ (%) | 68.66 | 73.91 | 72.08 | 62.13 | 47.18 | 56.97 | 80.36 | 67.73 |

${R}_{predict}^{2}$ (%) | 56.94 | 72.18 | 68.15 | 60.04 | 38.83 | 40.65 | 78.87 | 66.30 |

$rms{e}_{in-sample}$ (cm^{2}/(5 years)) | 21.70 | 22.67 | 60.53 | 31.56 | 33.86 | 45.20 | 26.19 | 54.89 |

$rms{e}_{predict}$ (cm^{2}/(5 years)) | 25.44 | 23.41 | 64.65 | 32.42 | 36.43 | 53.09 | 27.17 | 56.10 |

$rms{e}_{in-sample}$ (%) | 73.31 | 63.93 | 53.44 | 61.97 | 53.03 | 58.70 | 52.33 | 78.04 |

$rms{e}_{predict}$ (%) | 85.94 | 66.01 | 57.07 | 63.65 | 57.06 | 68.94 | 54.28 | 79.76 |

robust residual $\sigma $ | 1.52 | 1.38 | 2.34 | 1.80 | 1.91 | 2.22 | 1.46 | 2.08 |

weights, smaller 1 (in %) | 22.57 | 20.93 | 20.48 | 20.29 | 19.54 | 20.31 | 19.59 | 20.36 |

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## Share and Cite

**MDPI and ACS Style**

Zell, J. Climate Sensitive Tree Growth Functions and the Role of Transformations. *Forests* **2018**, *9*, 382.
https://doi.org/10.3390/f9070382

**AMA Style**

Zell J. Climate Sensitive Tree Growth Functions and the Role of Transformations. *Forests*. 2018; 9(7):382.
https://doi.org/10.3390/f9070382

**Chicago/Turabian Style**

Zell, Jürgen. 2018. "Climate Sensitive Tree Growth Functions and the Role of Transformations" *Forests* 9, no. 7: 382.
https://doi.org/10.3390/f9070382