# Deriving Tree Size Distributions of Tropical Forests from Lidar

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}/normalized RMSE 18.8%/R² 0.76; 50 ha: 22.8 trees ha

^{−1}/6.2%/0.89). Estimates for smaller scales (1-ha to 0.04-ha) were reliably for forests with low height, dense canopy or low tree height heterogeneity. Estimates for the basal area were accurate at the 1-ha scale (RMSE 4.7 tree ha

^{−1}, bias 0.8 m² ha

^{−1}) but less accurate at smaller scales. Our methodology, further tested at additional sites, provides a useful approach to determine the tree size distribution of forests by integrating information on tree allometries.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site

- (a)
- h = (57.4 × d)/(0.43 + d),
- (b)
- cr = 9.08 × d
^{0.68} - (c)
- vertical tree crown length cl (m) is linearly related to the tree height by [23]cl = 0.4 × h.

^{0.6}

^{−2}(in flight swath overlaps) with a median of 19 m

^{−2}. The point cloud was terrain-normalized and thinned using random subsampling to obtain a near homogeneous point density of four returns per m². The latter was achieved by iterative subsampling to different point densities and inspection of the resulting density rasters until no further density differences from the flight pattern were visible. The thinned point cloud contained 19% of the returns from the original point cloud. For details on the instrumentation and processing of the lidar data, see Lobo and Dalling [47] (Materials and Methods). The lidar dataset is publicly available and can be requested from J.W. Dalling (see statement on data accessibility in [47]).

#### 2.2. Derivation of Leaf Area Profiles from lidar

_{(i−1)},h

_{i}) of constant width Δh = 1 m is a result of (a) the tree crown density (LAD) at which lidar signals could get reflected and (b) the probability (W) that the lidar signal could penetrate into the respective height layer i (i.e., has not already been reflected in the layers above):

_{i}= l × LAD

_{i}× W

_{i}

_{i}in (m

^{-}²) is then defined as

_{ik}yields the number of lidar points with the return height z

_{k}(m) in the respective height layer i. To compute the probability W, we used the Beer–Lambert law of light transmission and extinction [48].

_{j}is the cumulative leaf area density (m²/m³) from the top of the forest (j = n) down to the height layer j = i+1. The parameter k in Equation (8) represents the average light extinction coefficient (k = 0.2 for near-infrared or infrared signals with wavelengths of 600–1400 µm) [48].

^{lidar}) via

_{n}= 0 results in W

_{n}= 1). We did not include ground returns in our approach and considered only height layers ≥ 3 m. The values of the parameters k and l are relevant for the derived leaf area profile, but had only minor influence on the estimation of the stem diameter distribution in our study (Figure A1).

#### 2.3. ‘Leaf–Tree Matrix’ of the Tree Geometry Model

_{ij}) with i,j = 1, …, n and the number of trees for each stem diameter class i by the vector N = (N

_{i})

_{i=1,...,n}(with n being the number of stem diameter classes, here n = 55). Multiplying the leaf–tree matrix F with the vector N results in the leaf area profile of a forest plot (LAD in m²/m³), in which we normalize by the plot area A (m²) and height layer width Δh (m)

^{lidar}of Equation (9)), which was used for the LAD (of Equation (10)). Rearranging Equation (10) yields the stem diameter distribution (unknown vector N).

#### 2.4. Linear Equation Solving to Derive Forest Structure from lidar Profiles

^{lidar}to represent LAD). See Appendix A Table A1 for a comparison of results between the direct and numerical calculation and Appendix B for details on the algorithm of numerical calculation.

#### 2.5. Analysis and Statistics of Results

**,**using the range of censused stem numbers across all stem diameter classes to normalize)

## 3. Results

#### 3.1. Stem Diameter Distribution Derived at the 50-ha Scale

#### 3.2. Small-Scale Derivations of Stem Diameter Distributions

^{lidar}, r = −0.19 to −0.31, Table A3) and total tree density (stem diameter d ≥ 1 cm, r = 0.12 to 0.27, Table A3, Figure 7).

## 4. Discussion

^{−1}), but the bias increased at smaller scales (−8.0 m² ha

^{−1}at 0.25-ha scale). We identified the most decisive factors for a good estimation of stem diameter distributions at smaller spatial scales as (a) the mean profile height (WMPH), (b) tree density, and (c) forest heterogeneity (in terms of tree height).

#### 4.1. Strengths and Limitations of the Presented Approach

#### 4.2. Future Applications and Challenges

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Sensitivity of the parameters k and l on the accuracy of estimating the stem diameter distribution (50-ha scale). (

**a**) The R² values, (

**b**) the nRMSE (%) values, (

**c**) the difference of the field-derived basal area (30.1 m²/ha) to lidar-derived estimates, and (

**d**) the difference of field-observed tree density (447.3 trees per ha) to lidar-derived estimates are shown for different combinations of parameter values (default values are k = 0.2 and l = 1). The quality of the values is indicated by gradients from green (good quality) to blue and red (lower quality). White cells demonstrate that no solution was possible due to infinite values in the leaf area profile derived from lidar (using the respective parameter combination). (

**e**) The estimated and observed stem diameter distributions compared for selected parameter combinations of k and l.

**Figure A2.**Visualization of the leaf–tree matrix and the resulting comparison of the measured tree numbers per stem diameter class (from inventory) with the estimated tree numbers (from lidar) for different sensitivity analysis. Upper graphics: each column in the matrix reflects the leaf area contribution of one tree to different height layers. Different columns represent different stem diameter classes. The leaf area values are shown using a color gradient from light green (low leaf area) to dark green (high leaf area) (white represents no leaf area). Lower graphics: green points show the estimated tree numbers from lidar for each stem diameter class and the grey line shows the observed stem numbers in the inventory. Values of R² and RMSE are given (see Table 3 for further results and a description of changed model assumptions in each scenario).

**Table A1.**Results of different calculations (direct: calculation of stem numbers N from Equation 11, numerical backward: numerical solving of Equation 10 for stem numbers N, see methods and Appendix B) to estimate the stem diameter distribution from lidar measurements (50-ha scale)

Method | Direct Calculation | Numerical Backward Solving |
---|---|---|

Regression slope | 1.22 | 1.24 |

Coefficient of determination R² | 0.88 | 0.89 |

Number of stem classes with negative predictions (%) | 26.9 | 0 |

RMSE (trees/ha) | 29.6 | 22.8 |

nRMSE (%) | 8.1 | 6.2 |

**Figure A3.**Evaluation of uncertainties in the lidar-derived tree density (in trees per ha) and lidar-derived basal area (in m²/ha) for three spatial scales: (

**a**) 50 ha, (

**b**) 1 ha, and (

**c**) 0.04 ha. Uncertainties are calculated as the absolute difference of stem numbers or basal area (logarithmic y-axes) per stem diameter class (x-axes) between the results from lidar and field data (points show the mean and the solid line shows the range from minimum to maximum). Note that zero differences are not displayed.

**Figure A4.**Spatial maps and histograms of the statistics on RMSE (trees/ha), nRMSE (%), and regression slope for the estimation of stem diameter distributions from lidar, (

**a**–

**c**) at the 1-ha scale, (

**d**–

**f**) at the 0.25-ha scale, and (

**g**–

**i**) at the 0.04-ha scale at BCI. Green colors denote the optimal values (note different value ranges for the different spatial scales and statistics). Black vertical lines in the histograms show the median.

**Table A2.**Comparison of the forest attributes derived from lidar and from the forest inventory for different plot sizes. Results are shown for the regression analysis (slope and R²), RMSE and nRMSE (overall and for three different categories of small, mid-sized and large trees grouped according to the stem diameter d) and aggregated forest attributes (tree density and basal area, for stem diameter d ≥ 10 cm). For each plot size, the mean ± standard deviation (and in brackets, the minimum and maximum) of the respective attribute is given. The plot size is given in ha with the number of plots and plot dimension in m × m. Forest attributes are compared between lidar and inventory also in terms of the bias, RMSE, and nRMSE (%) (see methods for details).

Plot Size (ha) | 25 | 5 | |
---|---|---|---|

Side length (m) in x-direction | 500 | 200 | |

Side length (m) in y-direction | 500 | 250 | |

Number of plots | 2 | 10 | |

Overall quality | Regression slope | 1.17 ± 0.05 (1.14, 1.21) | 1.0 ± 0.1 (0.8, 1.1) |

R² | 0.92 ± 0.005 (0.91, 0.92) | 0.84 ± 0.09 (0.65, 0.94) | |

RMSE (trees/ha) | 25.4 ± 2.3 (23.8, 27.0) | 34.1 ± 11.5 (17.1, 50.8) | |

nRMSE (%) | 7.0 ± 1.1 (6.2, 7.7) | 9.4 ± 3.3 (4.9, 14.1) | |

RMSE per size group | Small trees ^{1} | 596.2 ± 192.2 (460.3, 732.1) | 622.6 ± 240.2 (402.2, 1191.5) |

Mid-sized trees ^{2} | 36.0 ± 1.4 (35.0, 37.0) | 45.9 ± 7.3 (36.8, 55.0) | |

Large trees ^{3} | 2.1 ± 0.6 (1.6, 2.5) | 3.3 ± 1.4 (1.5, 5.1) | |

Tree density ^{4} (trees/ha) | Inventory-based | 447.3 ± 3.0 (445.2, 449.4) | 447.3 ± 25.1 (420.0, 489.4) |

lidar-derived | 562.3 ± 29.4 (541.5, 583.1) | 561.4 ± 65.4 (476.2, 715.4) | |

bias | −115.0 | −114.1 | |

RMSE | 116.5 | 128.3 | |

nRMSE (%) | 26.1 | 28.7 | |

Basal area ^{4} (m²/ha) | Inventory-based | 30.1 ± 2.4 (28.4, 31.8) | 30.1 ± 2.8 (25.5, 34.7) |

lidar-derived | 24.3 ± 1.3 (23.4, 25.2) | 25.3 ± 3.4 (19.6, 31.8) | |

bias | 5.8 | 4.7 | |

RMSE | 5.8 | 4.9 | |

nRMSE (%) | 19.4 | 16.4 |

^{1}Stem diameter classes d ≤ 10 cm.

^{2}Stem diameter classes 10 cm < d ≤ 50 cm.

^{3}Stem diameter classes d > 50 cm.

^{4}Stem diameter classes d ≥ 10 cm.

**Figure A5.**Spatial maps of the difference in (

**a**) tree density (trees/ha) and (

**b**) basal area (m²/ha) between lidar and inventory and respective histograms on the example of the 1-ha scale. Green colors denote the optimal values.

**Table A3.**Correlation coefficients (Spearman’s r) of R² values (of the compared lidar-derived and inventoried stem diameter distribution) correlated with other statistical measures, lidar data measures, and forest structure attributes for different plot sizes (1, 0.25, and 0.04 ha). See methods and Table 1 for details.

Category | Measure | Spatial Scale | ||
---|---|---|---|---|

1 ha | 0.25 ha | 0.04 ha | ||

Statistical measures | Regression slope | 0.63 | 0.54 | 0.73 |

RMSE | 0.09 | −0.15 | −0.20 | |

nRMSE (%) | 0.05 | −0.19 | −0.18 | |

lidar data | Profile height—median | −0.06 | −0.09 | −0.06 |

Profile height—maximum | −0.03 | −0.10 | −0.05 | |

Profile height—variance | −0.05 | −0.09 | −0.05 | |

Leaf area density—median | −0.13 | −0.03 | −0.01 | |

Leaf area density—maximum | 0.17 | −0.03 | −0.03 | |

Leaf area density—variance | 0.21 | 0 | −0.04 | |

WMPH | −0.31 | −0.25 | −0.19 | |

WVPH | −0.11 | 0.03 | 0.08 | |

Number of lidar returns | −0.12 | 0.09 | −0.04 | |

Forest inventory | Basal area (m²/ha) | −0.15 | −0.14 | −0.09 |

(for stem diameter d ≥ 10 cm) | −0.15 | −0.16 | −0.10 | |

Tree density (ha^{−1}) | 0.12 | 0.27 | 0.21 | |

(for stem diameter d ≥ 10 cm) | 0.17 | −0.02 | −0.06 | |

Standard deviation of tree height (m) | −0.16 | −0.28 | −0.19 | |

Median tree height (m) | 0.09 | −0.10 | −0.09 |

## Appendix B

#### Numerical Backward Calculation to Solve the Linear Equation System (Equation (10))

^{lidar}derived from the lidar observations (see methods). We then define L = A × LAD

^{lidar}as the leaf area across all height layers, with A as the plot area in m².

- If the leaf area L
_{i}in height layer i is larger than the corresponding entry f_{ii}in the leaf–tree matrix, we calculate the number of stems in the corresponding stem diameter class by N_{i}= ⎣L_{i}/(f_{ii})⎦.- 1a.
- If tolerance is not yet reached, i.e., (L
_{i}– N × f_{ii}) > ε × L_{i}then N_{i}= N_{i}+ 1. - 1b.
- The leaf area corresponding to the calculated number of trees N
_{i}in the respective stem diameter class i is then subtracted from all height layers j (below layer i) in which those trees also reach in with their crown: L_{j}= L_{j}− N_{i}× f_{ij}.

- If the leaf area L
_{i}in height layer i is lower than the corresponding entry f_{ii}in the leaf–tree matrix F, we set the number of stems in stem diameter class i to zero (i.e., N_{i}= 0).

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**Figure 1.**Combination of light detection and ranging (lidar) measurements with the tree geometry model to estimate the stem diameter distribution of a forest. The starting point is (

**a**) the measured lidar point cloud (here shown for an exemplary 1 ha subplot of Barro Colorado Island (BCI)). Points are colored according to their height (with a gradient from the top with blue colors to the bottom with green colors). (

**b**) The vertical leaf area profile can be derived from the lidar point cloud by accounting for decreasing lidar returns with decreasing height due to the tree leaf density (ground returns of height < 3 m are not shown). The vertical leaf area profile is then combined with the tree geometry model illustrated in (

**c**). By calculating how many trees of specific size (illustrated by different shadings) can contribute tree crown leaves to the lidar -derived leaf area profile, we can estimate (

**d**) the number of trees (per ha) in specific stem diameter classes (here, logarithmic y-axis, 50 ha plot of BCI, 20 cm diameter class width). See methods for details.

**Figure 2.**Visualization of the leaf–tree matrix (

**left**) and an enlarged cut-out (

**right**). Each column in the matrix reflects the leaf area contribution of one tree to different height layers. Columns represent different stem diameter classes. Colors in the matrix show leaf area values with a gradient from light green (low values) to dark green (high values) (white indicating no leaf area). On the right, an exemplary tree of 10 cm stem diameter is highlighted in the cut-out with its crown leaf area distributed to four height layers. Note the logarithmic x-axis of the stem diameter (cm).

**Figure 3.**Comparison of the stem diameter distribution derived from lidar with field data at BCI (50 ha). (

**a**) The tree number per stem diameter class derived from lidar (green points) and field inventory (grey line) of BCI on log–log axes (see methods for details). (

**b**) 1:1 plot of the comparison on log–log axes including a linear regression line (blue line) for evaluation. Grey dots represent the logarithmic tree numbers for each stem diameter class comparing those derived from lidar (y-axis) and those censused in the inventory (x-axis). For the root mean square error (RMSE) (22.8 trees per ha) and normalized RMSE (nRMSE) (6.2%) values, we refer to the methods and Table 3.

**Figure 4.**Stem diameter distributions derived from lidar for scales of 1, 0.25, and 0.04 ha. (

**a**) An exemplary 1-ha plot of the BCI forest is illustrated by comparing the stem diameter distribution from inventory (grey line) with that derived from lidar (green points). (

**b**) Map of BCI divided in 1-ha subplots and colored according to R² values. Values close to one (green) show a good estimation. The exemplary plot shown in (

**a**) is highlighted with a black border. (

**c**) Histogram of R² values for all subplots of 1 ha (the vertical line shows the median). Similar analysis is shown in (

**d**–

**f**) for the 0.25-ha scale and in (

**g**–

**i**) for the 0.04-ha scale. An analysis of the regression slope, RMSE and nRMSE values is shown in Table 3 and Figure A4.

**Figure 5.**Scaling relations between plot size and goodness-of-fit measures. (

**a**) The RMSE between lidar-derived and inventory-based stem numbers of subplots is shown (mean: black dots, standard deviation: grey polygon) with increasing plot size (ha). Please note the log–log axes. A linear regression (black line, based on black dots) provides a slope and R² of the respective scaling relation. A similar analysis is given for (

**b**) nRMSE (%), (

**c**) the regression slope of lidar-derived and inventory-based stem numbers, and (

**d**) coefficient of determination R². Please note the different sample sizes for the different spatial scales (see also Table 3 and Table A2 for details).

**Figure 6.**Comparison of lidar-derived and inventory-based (

**a**–

**c**) tree density (N) and (

**d**–

**f**) basal area (BA) for the (

**a**,

**d**) 1-ha scale, (

**b**,

**e**) 0.25-ha scale, and (

**c**,

**f**) 0.04-ha scale. Each black dot represents one subplot of the respective spatial scale. Dotted lines denote the 1:1 line. Blue lines show the linear regression line (with its equation and R² shown in each panel). Values on the bias, RMSE, and nRMSE are provided in Table 3. Please note the different axis ranges and sample sizes for the different spatial scales (see also Table 3).

**Figure 7.**Correlations of R² with characteristics of the lidar profile, forest attributes, and profile differences. In (

**a**) correlations of R² values and the mean profile height (WMPH) is shown at the 0.04-ha (left), 0.25-ha (middle) and 1-ha scale (right). Each black dot represents a forest plot, and the blue dashed lines reflect the optimal values for R² (close to one). The correlation coefficients are displayed below (Spearman’s r). In (

**b**,

**c**) the correlation coefficients are shown for each plot size: in (

**b**) lidar-derived attributes (black: profile height WMPH, green: variance of leaf density) and in (

**c**) forest attributes (black: tree density for d ≥ 1 cm, green: standard deviation of tree height). Grey dashed vertical lines show no dependence (r = 0).

**Table 1.**List of attributes used for the correlations with R² (from the comparison of the estimated stem diameter distribution with field data). Columns denote the data used, a description of the attribute, its unit and the different statistical calculations applied.

Data Source | Attribute | Unit | Calculations |
---|---|---|---|

lidar data | Profile height | m | Maximum, Median, Variance |

Leaf area density | m²/m³ | Maximum, Median, Variance | |

Number of lidar returns | 1/m^{2} | - | |

profile height weighted by leaf area | m | Median (WMPH ^{1}), Variance (WVPH ^{2}) | |

Forest inventory | Basal area ^{3,4} | m²/ha | Sum |

Tree density ^{3,4} | 1/m^{2} | Sum | |

Tree height ^{3} | m | Median, Standard deviation |

^{1}Median height of the vertical leaf profile derived from lidar (weighted by the derived leaf area density LAD

^{lidar}per height layer).

^{2}Variance of height of the vertical leaf profile derived from lidar (weighted by the derived leaf area density LAD

^{lidar}per height layer).

^{3}Including trees of stem diameter d ≥ 1 cm.

^{4}Including trees of stem diameter d ≥ 10 cm.

**Table 2.**Sensitivity analysis of different model assumptions (see also Figure A2). The changed model assumptions for each scenario are written in bold.

Sensitivity Scenario | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Crown shape | Sphere | Cylinder | Ellipsoid | Ellipsoid |

Height allometry | Asymptotic | Asymptotic | Asymptotic | Power law |

Crown length | Crown radius | 0.4 × height | 0.4 × height | 0.4 × height |

Crown leaf density | 0.44 m²/m³ | 0.44 m²/m³ | 1 m²/m³ | 0.44 m²/m³ |

Regression slope | 1.22 | 1.29 | 1.21 | 1.1 |

R² | 0.89 | 0.98 | 0.92 | 0.89 |

RMSE (trees/ha) | 32.0 | 4.5 | 15.0 | 15.3 |

nRMSE (%) | 8.8 | 1.2 | 4.1 | 3.8 |

**Table 3.**Comparison of forest attributes derived from lidar and from the forest inventory for different plot sizes. Results are shown for the regression analysis (slope and R²), RMSE and nRMSE (overall and for three different categories of small, mid-sized and large trees grouped according to stem diameter d) and aggregated forest attributes (tree density and basal area, for stem diameter d ≥ 10 cm). For each plot size, the mean ± standard deviation (and in brackets, minimum and maximum) of the respective attribute is given. The plot size is given in ha with the number of plots and plot dimension in m × m. Forest attributes are compared between lidar and inventory also in terms of the bias, RMSE, and nRMSE (%) (see methods for details). See Table A2 for additionally analyzed plot sizes.

Plot Size (ha) | 50 | 1 | 0.25 | 0.04 | |
---|---|---|---|---|---|

Side length (m) in x-direction | 1000 | 100 | 50 | 20 | |

Side length (m) in y-direction | 500 | 100 | 50 | 20 | |

Number of plots | 1 | 50 | 200 | 1250 | |

Overall quality | Regression slope | 1.24 | 0.9 ± 0.14 (0.66, 1.23) | 0.76 ± 0.2 (0.11, 1.3) | 0.55 ± 0.38 (−0.94, 2.17) |

R² | 0.89 | 0.76 ± 0.13 (0.38, 0.95) | 0.67 ± 0.17 (0.13, 0.94) | 0.44 ± 0.27 (0.00005, 1) | |

RMSE (trees/ha) | 22.8 | 67.6 ± 45.1 (15.2, 187.4) | 118.1 ± 77.6 (13.4, 385.6) | 219.2 ± 160.4 (17.7, 1074.6) | |

nRMSE (%) | 6.2 | 18.8 ± 13.2 (4.2, 55.0) | 33.2 ± 21.7 (3.0, 105.7) | 70.9 ± 62.0 (3.2, 590.1) | |

RMSE per size group (trees/ha) | Small trees ^{1} | 590.5 | 719.5 ± 379.6 (188.6, 1844.9) | 796.8 ± 492.6 (153.6, 3028.1) | 902.2 ± 751.6 (128.4, 5423.6) |

Mid-sized trees ^{2} | 39.4 | 77.1 ± 30.5 (28.0, 175.3) | 112.9 ± 43.1 (39.9, 267.6) | 153.9 ± 73.9 (33.5, 606.7) | |

Large trees ^{3} | 1.2 | 5.1 ± 2.5 (2.2, 15.9) | 9.1 ± 5.5 (2.4, 36.1) | 26.2 ± 21.2 (0, 230.3) | |

Tree density ^{4} (trees/ha) | Inventory-based | 447.3 | 447.3 ± 46.1 (353, 597) | 447.3 ± 60.4 (300, 648) | 447.3 ± 115.7 (175, 1025) |

lidar-derived | 516.7 | 615.5 ± 159.3 (364, 1320) | 758.9 ± 249.2 (316, 1864) | 992.6 ± 402.5 (150, 3125) | |

bias | −69.4 | −168.2 | −311.6 | −545.3 | |

RMSE | 69.4 | 228.4 | 394.4 | 677.8 | |

nRMSE (%) | 15.5 | 51.1 | 88.2 | 151.5 | |

Basal area ^{4} (m²/ha) | Inventory-based | 30.1 | 30.1 ± 5.1 (20.4, 45.8) | 30.1 ± 8.3 (15.5, 67.9) | 30.1 ± 20.9 (3.5, 206.1) |

lidar-derived | 23.6 | 29.3 ± 6.9 (18.8, 58.1) | 38.1 ± 15.0 (9.5, 163.4) | 59.0 ± 45.0 (4.7, 762.5) | |

bias | 6.5 | 0.8 | −8.0 | −28.9 | |

RMSE | 6.5 | 4.7 | 14.3 | 49.8 | |

nRMSE (%) | 21.6 | 15.7 | 47.4 | 165.3 |

^{1}Stem diameter classes d ≤ 10 cm.

^{2}Stem diameter classes 10 cm < d ≤ 50 cm.

^{3}Stem diameter classes d > 50 cm.

^{4}Stem diameter classes d ≥ 10 cm.

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**MDPI and ACS Style**

Taubert, F.; Fischer, R.; Knapp, N.; Huth, A.
Deriving Tree Size Distributions of Tropical Forests from Lidar. *Remote Sens.* **2021**, *13*, 131.
https://doi.org/10.3390/rs13010131

**AMA Style**

Taubert F, Fischer R, Knapp N, Huth A.
Deriving Tree Size Distributions of Tropical Forests from Lidar. *Remote Sensing*. 2021; 13(1):131.
https://doi.org/10.3390/rs13010131

**Chicago/Turabian Style**

Taubert, Franziska, Rico Fischer, Nikolai Knapp, and Andreas Huth.
2021. "Deriving Tree Size Distributions of Tropical Forests from Lidar" *Remote Sensing* 13, no. 1: 131.
https://doi.org/10.3390/rs13010131