# Analysis of Height Growth Suggests Moderate Growth of Tilia cordata and Acer platanoides at the Native Hemiboreal Stands in Latvia

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Study Area and Measurements

^{2}plots were established and re-measured (height of each tree and age) every fifth year. Tree height was measured using a Vertex clinometer (Haglöf Sweden AB, Långsele, Sweden) with an accuracy of 0.1 m, but, for age determination, increment cores from two to three dominant trees at a height of 1.3 were sampled with a Pressler increment corer. In this study, only the plots with forestry land-use type and Norway maple or small-leaved lime as the canopy species were selected. The selected 214 plots (86 for lime and 145 for maple; both species can coexist in one plot) were equally distributed across Latvia (Figure 1). The average stand age estimates of the maple and lime were 39 (from 4 to 120) and 47.5 years (from 3 to 103), respectively (Table 1). The selected stands generally represented mesotrophic and oligotrophic stands on freely draining mineral soils; some stands (<10%) grew on drained peat or oligotrophic mineral soils.

#### 2.2. Model Fitting

_{1}was the height at age A

_{1}, and H

_{2}was the height at age A

_{2}(in metres and years, respectively). The models were developed for dominant heights above 1.3 m. The mean age, when the studied trees had reached the breast height, was assumed to be three years, irrespective of species.

^{2}), root-mean-square error (RMSE) and the Akaike information criterion (AIC) were used. The estimated models were graphically compared to the yield tables from Lithuania [42], Germany [43] and Poland [44] for small-leaved lime and from Romania [45] for Norway maple. Data analysis was performed in R v. 4.2.2 [46].

**Table 2.**The generalised algebraic difference approach (GADA) models tested in height growth model development for small-leaved lime and Norway maple.

Base Model | $\mathbf{Solution}\mathbf{for}\mathbf{Theoretical}\mathbf{Variable}\mathsf{\chi}$ | Generalised Algebraic Difference Approach Model |
---|---|---|

Chapman–Richards: $H=1.3+{a}_{1}{\xb7\left[1-exp\left(-{a}_{2}\xb7A\right)\right]}^{{a}_{3}}$ | $\chi =\frac{1}{2}\xb7\left[\omega +\sqrt{{\omega}^{2}-4{\xb7b}_{3}\xb7\phi}\right]$ with $\omega =\left(\mathit{ln}{H}_{1}-{b}_{2}\xb7\phi \right)$ and $\phi =\mathit{ln}\left(1-exp\left[-{b}_{1}{\xb7A}_{1}\right]\right)$ | [47] ${H}_{2}=1.3+\left({H}_{1}-1.3\right){\left(\frac{1-exp\left[-{b}_{1}{\xb7A}_{2}\right]}{1-exp\left[-{b}_{1}{\xb7A}_{1}\right]}\right)}^{\left({b}_{2}+\frac{{b}_{3}}{\chi}\right)}$ |

Hossfeld $H=1.3+\frac{{a}_{1}}{{1+a}_{2}\xb7{A}^{{-a}_{3}}}$ | $\chi =\frac{{H}_{1}-{b}_{1}}{1-{b}_{2}\xb7{H}_{1}\xb7{{A}_{1}}^{-{b}_{3}}}$ | [30] ${H}_{2}=1.3+\frac{{b}_{1}+\chi}{1+{b}_{2}\xb7\chi \xb7{{A}_{2}}^{-{b}_{3}}}$ |

Hossfeld I $H=1.3+\frac{{A}^{2}}{{{a}_{1}+a}_{2}\xb7A+{a}_{3}{\xb7A}^{2}}$ | $\chi =\frac{{A}_{1}^{2}\xb7\left(1-{b}_{1}\xb7{H}_{1}\right)-{b}_{1}\xb7{H}_{1}}{{A}_{1}\xb7{H}_{1}\xb7\left(1+{b}_{2}\xb7{A}_{1}\right)}$ | [34] ${H}_{2}=1.3+\frac{{A}_{2}^{2}}{{b}_{1}\xb7\left(1+{A}_{2}^{2}\right)+\chi {\xb7A}_{2}\xb7\left(1+{b}_{2}\xb7{A}_{2}\right)}$ |

Hossfeld IV $H=1.3+\frac{{A}^{{a}_{1}}}{{a}_{2}+{a}_{3}{\xb7A}^{{a}_{1}}}$ | $X=\frac{\frac{{A}_{1}^{{b}_{1}}}{{H}_{1}-1.3}-{b}_{2}}{{b}_{3}+{A}_{1}^{{b}_{1}}}$ | [47] ${H}_{2}=1.3+\frac{{A}_{2}^{{b}_{1}}}{{b}_{2}+{b}_{3}\xb7\chi +\chi {\xb7A}_{2}^{{b}_{1}}}$ |

Sloboda $H=1.3+{a}_{1}\xb7\mathrm{e}\mathrm{x}\mathrm{p}\left[{-a}_{2}\xb7exp\left(\frac{{a}_{3}}{\left({a}_{4}-1\right)\xb7{A}^{\left({a}_{4}-1\right)}}\right)\right]$ | $X=\frac{ln\left(\frac{{H}_{1}}{{b}_{1}}\right)}{exp\left(\frac{{b}_{2}}{\left({b}_{3}-1\right)\xb7{A}_{1}^{\left({b}_{3}-1\right)}}\right)}$ | [48] ${H}_{2}={b}_{1}\xb7{\left(\frac{{H}_{1}}{{b}_{1}}\right)}^{exp\left(\frac{{b}_{2}}{\left({b}_{3}-1\right)\xb7{A}_{2}^{\left({b}_{3}-1\right)}}-\frac{{b}_{2}}{\left({b}_{3}-1\right)\xb7{A}_{1}^{\left({b}_{3}-1\right)}}\right)}$ |

Strand $H=1.3+{\left(\frac{A}{{a}_{1}+{a}_{2}\xb7A}\right)}^{{a}_{3}}$ | $X=\frac{{A}_{1}\xb7\left({H}_{1}^{-\frac{1}{{b}_{3}}}-{b}_{1}\right)}{1+{b}_{2}\xb7{A}_{1}}$ | [34] ${H}_{2}={\left(\frac{{A}_{2}}{\chi +{A}_{2}\xb7\left({b}_{1}+{b}_{2}\xb7\chi \right)}\right)}^{{b}_{3}}$ |

_{1}, a

_{2,}a

_{3}and a

_{4}are parameters in base models; b

_{1}, b

_{2}and b

_{3}are parameters in dynamic models; H

_{1}and H

_{2}are heights (in m) at breast high age A

_{1}and A

_{2}(in years), respectively.

## 3. Results

## 4. Discussion

^{2}, AIC in Table 3) and range of estimated parameters (Table 4) of the developed height growth models for lime and maple in Latvia were comparable with those for similar estimates [29,34]. The fit statistics, cross-validation (Table 3) and biological realism (Figure 2) indicated the Sloboda and Chapman–Richards models as the best for estimating the height of lime in Latvia. However, the Sloboda model apparently better corresponded to species characteristics, particularly the rapid height growth when young (15–25 years), which slows afterwards [20]. Although the curves of Chapman–Richards and Strand models for the highest site indices corresponded to fast-starter ecology and were comparable to European models [42,43,44], both models apparently exaggerated the dynamics for the top site index, adding to the uncertainty of prediction. The slow initial growth estimated for the lower site indices visible in Chapman–Richards, Sloboda and Strand models can, however, be explained by shade tolerance [51]. Also, this might be an artefact of competition for the lime growing in the understory of mixed stands [20]. The developed dominant height models showed rather poor conformity with others [42,43,44], indicating explicit bias. Marginal conformity was only with the highest site index trees from the Polish model. This confirms the importance of regional models for the accuracy of height growth prediction [27]. The estimated higher-age acceleration of height growth for the lower site indices (Figure 2) might be related to the ameliorating effects of climatic changes and eutrophication/nitrogen deposition, particularly in poor sites [52]. Such differences might also be related to an unbalanced coverage of data as well as different behaviours of the models (polymorphic with various or partially various asymptotes; [30]). Alternatively, this might be an artefact due to the underrepresentation of the data.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Location of the studied sites used for the development of height growth models for Norway maple and small-leaved lime in Latvia. Circles denote sites of Norway maple; triangles denote sites of small-leaved lime.

**Figure 2.**The dominant height models (black dashed lines) fitted to the observed data of 510 trees (grey lines, each line represents a single tree) for small-leaved lime and Norway maple; the model predictions are for 4 m site index intervals for the range 18–34 m at the base age of 100 years, which is common for hardwoods in the Eastern Baltic region.

**Figure 3.**The differences between small-leaved lime and maple dominant height predicted by the best-developed Sloboda and Hossfeld I models of Latvia and yield tables for small-leaved lime of Lithuania [42], Germany [43] and Poland [44] in addition to Norway maple of Romania [45], Sycamore of Denmark [50] and Germany [49] according to stand age and site index.

**Table 1.**The basic characteristics of small-leaved lime and Norway maple datasets used for the calibration and cross-validation of height growth models in Latvia based on the National Forest Inventory.

Variable (Unit) | Statistic | Small-Leaved Lime | Norway Maple | ||
---|---|---|---|---|---|

Calibration | Cross-Validation | Calibration | Cross-Validation | ||

Age (year) | Mean | 47.5 | 49.2 | 39.0 | 35.7 |

St. dev. | 22.7 | 23.7 | 24.8 | 19.6 | |

Min | 3 | 3 | 4 | 4 | |

Max | 103 | 103 | 122 | 120 | |

Height (m) | Mean | 19.5 | 19.3 | 17.1 | 17.0 |

St. dev. | 6.2 | 6.5 | 5.9 | 5.5 | |

Min | 3.5 | 4.4 | 3.9 | 5.3 | |

Max | 31.5 | 32.5 | 33.6 | 32.7 | |

Sample plots (-) | Number | 81 | 47 | 126 | 70 |

Trees (-) | Number | 197 | 87 | 224 | 92 |

Measurements of trees (-) | Number | 277 | 100 | 300 | 100 |

**Table 3.**Models fit statistics and cross-validation: mean residual (MR), root-mean-square error (RMSE), adjusted R

^{2}value and Akaike information criterion (AIC) for height growth model development of small-leaved lime and Norway maple in Latvia.

Species | Model | MR (m) | RMSE (m) | adj. R^{2} | AIC | |
---|---|---|---|---|---|---|

Fit statistic | Small-leaved lime (N = 277) | Chapman–Richards | 0.01 | 0.84 | 0.980 | −95.7 |

Hossfeld | 0.08 | 0.86 | 0.979 | −79.7 | ||

Hossfeld I | −0.04 | 0.86 | 0.980 | −83.5 | ||

Hossfeld IV | 0.06 | 0.86 | 0.979 | −80.0 | ||

Sloboda | 0.01 | 0.84 | 0.980 | −96.2 | ||

Strand | 0.02 | 0.84 | 0.980 | −97.3 | ||

Norway maple (N = 300) | Chapman–Richards | 0.01 | 0.94 | 0.984 | −37.1 | |

Hossfeld | 0.18 | 1.02 | 0.982 | 14.0 | ||

Hossfeld I | 0.08 | 0.99 | 0.983 | −5.5 | ||

Hossfeld IV | 0.10 | 1.01 | 0.982 | 8.1 | ||

Sloboda | 0.01 | 0.94 | 0.984 | −34.5 | ||

Strand | 0.07 | 0.99 | 0.983 | −6.8 | ||

Cross-validation | Small-leaved lime (N = 100) | Chapman–Richards | 0.04 | 0.95 | 0.976 | −8.8 |

Hossfeld | 0.15 | 0.96 | 0.977 | −6.9 | ||

Hossfeld I | 0.01 | 0.96 | 0.976 | −6.7 | ||

Hossfeld IV | 0.13 | 0.95 | 0.977 | −7.1 | ||

Sloboda | 0.05 | 0.94 | 0.976 | −9.3 | ||

Strand | 0.06 | 0.96 | 0.976 | −5.5 | ||

Norway maple (N = 100) | Chapman–Richards | −0.10 | 1.04 | 0.981 | 10.5 | |

Hossfeld | 0.06 | 1.08 | 0.979 | 16.8 | ||

Hossfeld I | −0.02 | 1.05 | 0.980 | 11.3 | ||

Hossfeld IV | −0.02 | 1.07 | 0.979 | 16.4 | ||

Sloboda | −0.10 | 1.04 | 0.981 | 9.6 | ||

Strand | −0.04 | 1.05 | 0.980 | 12.6 |

**Table 4.**Parameter estimates (b

_{1}, b

_{2}, b

_{3}) of the fitted dynamic height growth models of small-leaved lime and Norway maple in Latvia.

Model | Parameter | Small-Leaved Lime | Norway Maple | ||
---|---|---|---|---|---|

Estimate | Standard Error | Estimate | Standard Error | ||

Chapman–Richards | b_{1} | 0.0091 | 0.0026 | 0.0254 | 0.0025 |

b_{2} | −417.22 | 11.49 | −80.60 | 17.49 | |

b_{3} | 1585.68 | 3.01 | 284.27 | 60.35 | |

Hossfeld | b_{1} | 63.57 | 10.17 | 50.62 | 7.93 |

b_{2} | −44.32 | 0.0111 | −16.2958 | 0.0481 | |

b_{3} | 0.8537 | 0.0453 | 1.0402 | 0.0502 | |

Hossfeld I | b_{1} | 0.0283 | 0.0011 | 0.0255 | 0.0015 |

b_{2} | −0.0061 | 0.0004 | −0.0053 | 0.0005 | |

Hossfeld IV | b_{1} | 0.8405 | 0.0469 | 1.0338 | 0.0564 |

b_{2} | −1449.54 | 50.22 | −2451.73 | 94.81 | |

b_{3} | 970.38 | 75.08 | 1264.81 | 182.70 | |

Sloboda | b_{1} | 45.12 | 9.29 | 34.82 | 2.69 |

b_{2} | 0.1387 | 0.0212 | 0.1781 | 0.0248 | |

b_{3} | 0.5589 | 0.0688 | 0.5024 | 0.0597 | |

Strand | b_{1} | 0.0098 | 0.0046 | 0.0342 | 0.0125 |

b_{2} | −0.0063 | 0.0005 | −0.0054 | 0.0005 | |

b_{3} | 0.7799 | 0.0670 | 1.0769 | 0.1000 |

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## Share and Cite

**MDPI and ACS Style**

Matisone, I.; Šņepsts, G.; Kaupe, D.; Hein, S.; Rieksts-Riekstiņš, R.; Jansons, Ā.
Analysis of Height Growth Suggests Moderate Growth of *Tilia cordata* and *Acer platanoides* at the Native Hemiboreal Stands in Latvia. *Forests* **2024**, *15*, 7.
https://doi.org/10.3390/f15010007

**AMA Style**

Matisone I, Šņepsts G, Kaupe D, Hein S, Rieksts-Riekstiņš R, Jansons Ā.
Analysis of Height Growth Suggests Moderate Growth of *Tilia cordata* and *Acer platanoides* at the Native Hemiboreal Stands in Latvia. *Forests*. 2024; 15(1):7.
https://doi.org/10.3390/f15010007

**Chicago/Turabian Style**

Matisone, Ilze, Guntars Šņepsts, Dārta Kaupe, Sebastian Hein, Raitis Rieksts-Riekstiņš, and Āris Jansons.
2024. "Analysis of Height Growth Suggests Moderate Growth of *Tilia cordata* and *Acer platanoides* at the Native Hemiboreal Stands in Latvia" *Forests* 15, no. 1: 7.
https://doi.org/10.3390/f15010007