Introduction

Agroforestry systems (AFS) have been flagged as a possible contribution to climate change mitigation and adaption. Compared to conventional agricultural systems, AFS provide a higher carbon sequestration potential (Montagnini and Nair 2004; Zomer et al. 2016; Abbas et al. 2017). They serve as in-situ carbon sinks, but by using the produced timber in long-lived products, long-term ex-situ carbon storage can be achieved as well (Jose 2009). Additionally, woody components in AFS enhance the structural heterogeneity of the agricultural landscape, leading to higher habitat heterogeneity which in turn promotes biodiversity (Benton et al. 2003). In several studies, AFS have been shown to increase biodiversity compared to conventional agricultural systems (Palma et al. 2007; Torralba et al. 2016; Udawatta et al. 2019; Mupepele et al. 2021). By providing numerous ecosystem services, such as habitat structures and food for pollinators, erosion control, or soil and water quality improvement, trees and shrubs can benefit the agricultural component as well (Jose 2009; Wilson and Lovell 2016).

Despite the broad range of potential benefits and the long history of AFS (Mosquera-Losada et al. 2012), knowledge about the woody component in AFS is still lacking. Often, the research focus is on the agricultural component which is perceived to be more relevant for the cultivators. In particular, quantification of biomass and carbon stocks of the woody components is scarce, as its acquisition is both difficult and laborious. As opposed to forest carbon stocks, which can be estimated with well-established allometric functions, for trees outside forests (TOF) these functions are typically lacking. Forest-specific allometric functions, however, cannot simply be transferred to TOF, as the growth conditions and the management are very distinct (Nair 2012; Schnell et al. 2015; Rötzer et al. 2021), leading to different crown dimensions and allometric relationships (Pretzsch et al. 2015).

Depending on economic and political incentives, AFS may be optimized towards different goals, such as timber production (Balandier and Dupraz 1998; Morhart et al. 2014), carbon sequestration (Nair et al. 2009), crop production (Nair 1993), animal welfare (Mele et al. 2019), or provision of pollinator habitats (Bentrup et al. 2019). To optimize AFS, knowledge about the individual components and inherent interactions is a paramount. Information on different tree species and their behavior in terms of dimensions and ecosystem services enables AFS design and guides management decisions. By measuring single tree volume and combining it with density and nutrient content, biomass as well as the amount of stored carbon and nutrients can be derived. While single tree dimensions are relevant to AFS design, quantification of carbon content is relevant for estimating carbon sequestration potential. Nutrient content is important when assessing nutrient loss upon tree harvesting or modelling nutrient cycling. Tree volume and biomass are commonly determined by destructive sampling, which is very laborious. To limit the necessary efforts, bark and wood volumes are rarely collected separately (Morhart et al. 2016). However, bark and wood are known to differ in their physical and chemical properties (Block et al. 2008). Therefore, neglecting the bark fraction as a separate compartment decreases the accuracy of the estimated carbon content (Neumann and Lawes 2021) and presumably also of the nutrient content.

An efficient and non-destructive alternative to manual data collection is stem and branch volume estimation based on remote sensing data. To efficiently estimate single tree above-ground volume, TLS (Terrestrial Laser Scanning) can be employed. TLS is a ground-based remote sensing technology for deriving very dense and highly accurate 3D point clouds of the surroundings based on laser distance measurements. In forest sciences, TLS has been used for nearly two decades for various purposes (Dassot et al. 2011; Calders et al. 2020). To extract single tree information from 3D point clouds, QSMs (Quantitative Structure Models) have been successfully used. QSMs are 3D tree structure models which are built from 3D point clouds using geometric primitives, such as cylinders or cones. They can be used to derive topologic, geometric, and volumetric data (Åkerblom et al. 2015; Åkerblom and Kaitaniemi 2021). Possible applications of QSMs include the derivation of forestry related parameters (Calders et al. 2015), quantification of tree biomass (Smith et al. 2014; Gonzalez de Tanago et al. 2018), tree growth (Sheppard et al. 2017) and modelling of shade cast (Rosskopf et al. 2017; Bohn Reckziegel et al. 2021), among others.

Common walnut (Juglans regia L.) is a particularly promising tree species for AFS as it is an important species for the production of high value timber, either in the form of veneer or for the production of furniture or gun stocks where the hard, dark wood is able to fetch a high price. The species is a rich source of non-wood forest products, its nuts are frequently grown for human consumption (Ciesla 2002; Rigo et al. 2016), and it has multiple applications in pharmaceutical, chemical and material industries (Pereira et al. 2007; Oliveira et al. 2008; Delaviz et al. 2017; Croitoru et al. 2019). Walnut trees display strong phototropism and are extremely light demanding. Seedlings are more tolerant to shade than mature trees, but poor form may result due to a prolonged underexposure to sufficient light (Mohni et al. 2009), likewise the species is sensitive to late frost (Rigo et al. 2016), for these reasons they are not common in closed forests. This could be an explanation for the lack of dendrological information on this species.

Although walnut trees have been previously studied using laser scanning, e.g. by Estornell et al. (2021), this study was limited to few structural parameters and trees within a small stem diameter range. Furthermore, the derived models were constructed in such a way that they cannot be used with field measurements. Therefore, our aim is to investigate single tree parameters of walnut trees grown in AFS for trees within a larger diameter range in more detail. To this end, we derived above-ground single tree parameters from leafless trees using 3D data gathered using TLS and QSMs. By combining these data with auxiliary data, we were able to obtain a broad set of parameters. Our research objectives were:

  1. (1)

    To develop allometric functions relating tree height and crown parameters (base height, diameter, projection area) to easily obtainable variables based on TLS data.

  2. (2)

    To develop allometric functions relating whole tree above-ground volume, biomass, carbon content and nutrient content to easily obtainable variables based on TLS and auxiliary data.

  3. (3)

    To contextualize the results in a practical agricultural context by applying the derived functions to an exemplary AFS.

The workflow we propose could serve as a blueprint for deriving the parameters presented for further tree species to facilitate AFS planning and management.

Methods

Study site

Data were collected at several locations in south-west Germany (Fig. 1a). The trees were located within a 20 km radius around the city of Freiburg (47°59′42″ N, 7°51′0″ E). Freiburg is located at an altitude of 278 m a.s.l. and is situated in the cool temperate climate zone. In the period from 1991 to 2020, the mean annual air temperature was 11 °C and the mean annual precipitation sum was 896 mm (Deutscher Wetterdienst 2022). In the vegetation period (April to September), the recorded long-term mean temperature is 16.5 °C and the precipitation sum 510 mm. A total of 65 trees with a DBH (diameter at breast height, measured at 1.3 m) of 1 cm to 77 cm were surveyed. They were chosen to cover a large range of tree dimensions. The trees were typically positioned in tree rows and hedges next to agricultural areas, including both agriculture and viniculture. Due to wide spacing between the trees, they displayed crown shapes similar to those of solitary trees (Fig. 1b,c). As the trees were not located on controlled research sites, moderate differences in management approach are assumed but cannot be retrospectively confirmed. Although we are unable to confirm this assumption, the crowns of the trees did not appear to have been pruned in the past.

Fig. 1
figure 1

Overview over the studied AFS. a Map of Germany (light blue), Baden-Württemberg (dark blue) and the scanning locations (dark circles), Geodata: © GeoBasis-DE / BKG (2023), b Photo of a solitary walnut tree on the site in Merzhausen, c Point cloud of the site in Merzhausen, coloured according to altitude a.s.l. with yellow representing high and purple low altitude. (Color figure online)

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Scanning & point cloud preprocessing

The scans were carried out under leaf-off conditions between December 2021 and February 2022. Scanning was undertaken using the RIEGL VZ-400i terrestrial laser scanner (RIEGL Laser Measurement Systems GmbH, Horn, Austria). A multi-point scanning procedure was applied meaning that each tree was scanned from multiple angles and distances to reduce occlusion effects and to ensure full coverage. As the RIEGL VZ-400i performs automatic on-board co-registration, the scans did not have to be manually co-registered. Resultant point clouds were filtered to remove isolated points (< 5 points in 10 cm neighborhood), points with a low reflectance (< -15 dB), points with a high deviation (> 15) and points located more than 50 m away from the scanner. If the point clouds appeared to be misaligned, a multi-station adjustment was conducted (see Demol et al. 2022). To increase the computational efficiency of the next steps, the filtered point clouds were down-sampled to one point per cm3. Single tree point clouds were extracted from the point clouds by manual segmentation using CloudCompare v2.11.3 (www.cloudcompare.org). Noisy single tree point clouds were additionally filtered using the "Statistical Outlier Removal" implemented in CloudCompare, which removes outliers based on neighborhood distance statistics. Points in cavities of trunks and branches were removed manually to facilitate a plausible QSM fitting.

Modelling QSMs

To derive volumetric information from point clouds, QSMs are frequently used. They have been tested in various studies and usually show good agreement with field-based measurements (Hackenberg et al. 2014; Calders et al. 2015; Gonzalez de Tanago et al. 2018). Here, QSMs were derived from the filtered single tree point clouds using TreeQSM v2.4.0 (Raumonen et al. 2013; Åkerblom 2020), which uses parametric cylinders to approximate the 3D tree structure (Fig. 2). The modelling process includes stochastic elements, leading to different outputs each time the model is run. For this reason, creating multiple models per tree is suggested to obtain robust estimates (Raumonen 2020). During the input parameter optimization, for each parameter combination five models per tree were derived. The input parameters were pre-selected based on the number of points per tree and whether the point clouds had large gaps on branch and trunk surfaces due to cavity removal. The best parameters for each tree were determined using the average distance between points and fitted cylinders. Using the optimized input parameters, for each tree 30 QSMs were modelled to obtain robust estimates for the derived parameters. In the following steps, the relevant parameters were derived for each model and then averaged. All of the subsequent data analyses were conducted using the statistical software environment R v4.1.3 (R Core Team 2022).

Fig. 2
figure 2

Photograph (left) and image of a QSM (right) from the same walnut tree with a height of 13.37 m. The QSM is colored according to the branch order of the fitted cylinders. (Color figure online)

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Deriving biomass, carbon and nutrients

In order to derive leafless above-ground biomass, carbon, and nutrient content from volume estimates, additional information on bark and wood proportions and properties is necessary. To obtain the required information, two walnut trees were felled in the same region and at the same time of the TLS data acquisition. From each tree, disks were extracted from both stem and branches.

To derive bark and wood volumes, the diameter dependent bark thickness was required. The over-bark radius of the disk from the core to the outside of the bark, as well as the bark thickness, were manually measured on 170 disks measuring on four radii aligned orthogonally to the cambium. For each disk, the root mean square values of the over-bark radius and bark thickness were doubled to obtain the average over-bark diameter and double bark thickness. The double bark thickness was modelled using a linear regression according to Eq. 1, which was originally introduced by Loetsch et al. (1973) and is an established formula for describing double bark thickness (Stängle and Dormann 2018; Jankovský et al. 2019). \(DBT\) is the double bark thickness (mm), \(DOB\) is the diameter over bark (mm), \({\beta }_{0}\) and \({\beta }_{1}\) are regression coefficients, and \(\varepsilon\) is a random error.

$$DBT = \beta_{0} + \beta_{1} \cdot DOB + \varepsilon$$
(1)

For the conversion of volume into biomass, the dry density of bark and wood were determined by water replacement, employing the Archimedean principle (Seifert and Seifert 2014). The densities were averaged over 20 bark and 20 wood samples which were dried using an Heraeus T6760 (Heraeus Group, Hanau, Germany) for more than 24 h at 105 °C until constant weight was reached. The basic density \({\rho }_{basic}\) was approximated for both wood and bark using dry density and volumetric shrinkage. In Eq. 2, \({m}_{dry}\) is the dry mass (kg), \({V}_{fresh}\) is the fresh volume (m3), \({V}_{dry}\) is the dry volume (m3), and \({\beta }_{v}=0.134\) is the volumetric shrinkage for walnut as provided by Bosshard (1982). The basic density values were then multiplied with the wood and bark volumes to derive their biomass.

$$\rho_{basic} = \frac{{m_{dry} }}{{V_{fresh} }} = \frac{{m_{dry} }}{{V_{dry} \cdot \left( {1 - \beta_{v} } \right)^{ - 1} }} = \frac{{m_{dry} }}{{V_{dry} }} \cdot \left( {1 - \beta_{v} } \right) = \rho_{dry} \cdot \left( {1 - \beta_{v} } \right)$$
(2)

To obtain the amount of stored carbon and nutrients per tree, carbon and nutrient content of the dry biomass a total of 12 mixed samples containing material from several disks were analyzed. For the mixed samples, material from different parts of the trees was combined. Two mixed samples each for wood and bark were taken from three different diameter classes: small (\(d\) ≤ 3 cm), medium (3 cm < \(d\)  ≤ 7 cm) and large (\(d\) > 7 cm) diameters. All samples were dried at 105 °C until a constant weight was reached. Then the samples were ground and dried for a second time. Half of the samples were used for the determination of carbon and nitrogen content, while the other half was used to analyze the nutrient content of the samples. Carbon and nitrogen content were determined by dry combustion using an Elementar Vario EL Cube. To determine nutrient concentration, the samples were digested in a microwave with 65% HNO3 and H2O2 (Bock 1972). The solutions were then analyzed using an Agilent 5800 ICP optical emission spectrometer.

Subsequently, the QSM data were combined with the auxiliary data. First, the bark thickness model was used to obtain estimates for the under-bark diameters of the cylinders based on their over-bark diameters. Estimates of wood and bark volume were derived for each cylinder by combining the over- and under-bark diameters with the cylinder lengths. Using these volumes and the average basic density values of wood and bark, the biomass of both compartments was derived for all cylinders. For each tree, the biomass was summed up for each of the diameter classes that were used in the nutrient analyses (\(d\) ≤ 3 cm; 3 cm < \(d\)  ≤ 7 cm; \(d\) > 7 cm) and multiplied with the respective relative carbon and nutrient content to obtain the total carbon and nutrient content of each diameter class.

Modelling allometric functions

All modelled allometric functions were based on the DBH as the predictor variable, as it is easy to measure with commonly available tools, such as a measuring tape or a caliper. Although additional parameters such as tree age and height might have improved the fits of the regressions (Magalhães and Seifert 2015; Mensah et al. 2016; Stängle and Dormann 2018), we accepted inferior fits in favor of better applicability since these further parameters are typically not easily accessible to AFS owners. The functional relationships between response variables and predictors were chosen based on several fit statistics (AIC, RMSE, R2), normality of the residuals, and the visual assessment of the model behavior in the extrapolation range up to a DBH of 100 cm. The residuals of all selected models and the bark thickness model were tested (\(p\) < 0.05) for normality using Kolmogorow-Smirnow and Anderson–Darling tests; they were all normally distributed. All predictors and predictor interactions had significant effects (\(p\) < 0.05) on the respective response variables.

The functions relating to total volume, biomass, carbon content, and nutrient content were modelled according to Eq. 3. \(Y\) is the respective dependent variable in m3 or kg, \(DBH\) is the diameter at breast height (cm), \({\beta }_{0}\) and \({\beta }_{1}\) are regression coefficients, and \(\varepsilon\) is a random error. This equation is equivalent to a power function and is commonly used for modelling volume and biomass using linear models (Adhikari 1995; Picard et al. 2012). By transforming dependent variables to the logarithmic scale, heteroscedasticity in the dataset can be accounted for, which is common in single tree volume and biomass models. To account for bias resulting from the back-transformation of predictions from the logarithmic scale to the linear scale, a bias correction factor (CF) was derived for each of the models (Baskerville 1972; Sprugel 1983). The calculation of each CF is presented in Eq. 4, with \(\widehat{\sigma }\) being the standard error of the estimate. After back-transforming predictions, they have to be multiplied with the CF of the respective regression to obtain bias-corrected predictions.

$$\ln \left( Y \right) = \beta_{0} + \beta_{1} \cdot \ln \left( {DBH} \right) + \varepsilon$$
(3)
$$CF = \exp \left( {\frac{{\hat{\sigma }^{2} }}{2}} \right)$$
(4)

To enable the estimation of economic value, the volume was separately estimated for the branch-free bole and the crown. The branch-free bole, which can often be sold for higher revenues, refers to the stem up to the crown base height, while the crown was defined as the remaining volume of the tree. The models for biomass, carbon and nutrient content were split into bark and wood fractions. To ensure additivity of the respective models as a requirement for plausible biomass estimations (Seifert and Seifert 2014; Magalhães and Seifert 2015), beta regressions were used to estimate the proportions of the respective compartments, as suggested by Douma and Weedon (2019). The beta regressions were implemented using the betareg package in R (Cribari-Neto and Zeileis 2010). For these regressions, the logit was used as link function and Eq. 5 was used to model the relationship between the respective fractions and the DBH. \({Y}_{\mathrm{frac}}\) is the fraction of the respective dependent variable, \(DBH\) denotes for the diameter at breast height (cm), \({\beta }_{0}\) to \({\beta }_{3}\) are regression coefficients, and \(\varepsilon\) is a random error. As confidence intervals (CIs) are not derivable using the employed package, they were bootstrapped in 1000 iterations.

$${\text{logit}}\left( {Y_{{{\text{frac}}}} } \right) = \beta_{0} + \beta_{1} \cdot \ln \left( {DBH} \right) + \beta_{2} \cdot DBH + \beta_{3} \cdot \ln \left( {DBH} \right) \cdot DBH + \varepsilon$$
(5)

The regression models for predicting the total values were combined with the models for the individual compartments by multiplying the bias-corrected total predictions with the predicted fractions. The CIs of the models were combined by multiplying the lower and upper ends of the CIs of the total predictions with the lower and upper ends of the bootstrapped CIs of the fractional predictions.

Further models describing the relationship of DBH with tree height (m), crown base height (m), crown diameter (m), and crown projection area (m2) were fitted according to Eq. 3. For each of the fitted models, the correction factor as described in Eq. 4 was also calculated.

Results

The DBH of the studied individuals ranged from 1 to 77 cm and tree height from 1.19 to 17.45 m (Fig. 3). Tree height did not increase linearly with DBH value, the model showed a continuously decreasing slope at higher DBH values. Crown base height (CBH) ranged from 0.07 to 3.09 m and showed the pattern of flattening at higher DBH values even more pronounced. However, the variation of the crown base height measurements around the fitted regression was more pronounced, which is underlined by the lower adjusted R2 value of the model. Crown projection areas ranged from 0.02 to 341.97 m2 and crown diameter ranged from 0.04 to 21.07 m, accordingly. The fitted regressions for the height and crown parameters are provided in Table. 1.

Fig. 3
figure 3

Single tree parameters as functions of the DBH. The points correspond to data derived from the QSMs, the lines to fitted regressions and the shaded areas to the 95% confidence intervals of the respective regression. The plots show the data and models on a tree height, b crown base height, c crown diameter, and d crown projection area. (Color figure online)

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Table 1 Model coefficients, correction factor (CF) and adjusted R2 for the regressions relating single tree height and crown parameters to DBH
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The total volume of the single trees reached a maximum value of 12.4 m3. In Table. 2, the model describing the total volume as a function of DBH is provided. The bark thickness model used for estimating bark and wood volume separately was defined as follows: \(DBT=3.4286+0.0794\bullet DOB\) (Adj. R2 = 0.9227). The root mean square diameters of the bark thickness sample disks ranged between 0.4  and 35.8 cm. Separating the total tree volume into bark and wood, the share of wood volume ranged from 34.91% to 76.42%. When separating whole tree volumes into branch-free bole and crown, the share of bole volume varied from 2.82% to 88.5% of the total volume. With increasing DBH, the proportion of bole volume to total volume decreased (Appendix Fig. 6). This model, however, exhibited only a moderate fit with a pseudo R2 of 0.4935 (Appendix Table. 5).

Table 2 Model coefficients, correction factor (CF) and adjusted R2 for the regressions describing single tree total volume, dry biomass, carbon content, and nutrient content
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On average, the derived basic density of the measured wood samples was 0.554 ± 0.006 g/cm3 (mean ± standard error), the basic density of the bark samples was 0.715 ± 0.007 g/cm3. The resulting dry biomass per tree ranged from 0.08 to 7,393.3 kg. When separating dry biomass into bark and wood biomass, the share of wood biomass ranged between 29.36% and 71.52%. The regression between DBH and total biomass is given in Table. 2 and the regression describing the proportion of wood volume to total volume is provided in Table. 5 (Appendix). With increasing DBH values, the proportion of bark biomass relative to the total above-ground dry biomass decreased (Fig. 4).

Fig. 4
figure 4

Single tree above-ground dry biomass as a function of DBH for whole trees (grey circles), their wood (light blue squares), and their bark (dark blue triangles). The points correspond to the data extracted from the QSMs. The lines show the regressions, while the shaded areas correspond to the 95% confidence intervals. The wood and bark regressions are the combined regressions of the total above-ground biomass and the wood and bark fraction of that biomass. (Color figure online)

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The analysis of nutrient content of wood and bark revealed a higher content of all nutrients in the bark, except for magnesium in the smallest diameter class (Table. 3). For most nutrients, with decreasing diameter of the wood samples an increase in nutrient content could be observed. The derived carbon per tree varied from 0.04 to 3,588.11 kg (equivalent to 0.15 kg to 13,168.36 kg CO2 based on a conversion factor of 3.67) and the proportion of carbon allocated in the wood fraction ranged from 30.74 to 71.76%. The models describing total carbon and nutrient content per tree can be found in Table. 2 and Fig. 7 (Appendix). The regressions relating the proportion of carbon and nutrient content of wood to the total content are shown in Table. 5 (Appendix) and Fig. 8 (Appendix). As DBH increased, the proportion of nutrients stored in the bark decreased and that in the wood increased.

Table 3 Relative nutrient content in mg per g dry biomass for wood and bark in three diameter classes
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Example AFS

To present the results in a practical context, an exemplary AFS was modeled. Its layout is shown in Fig. 5. The modeled AFS has a size of 1 ha with a length and width of 100 m each. The target DBH of the trees at the end of the rotation period was set to 60 cm. On the model plot, we included three tree strips consisting of five trees each. The planting distance of 20 m between the trees within a row was determined by predicting the crown diameter at the target DBH (see Fig. 3). Each tree strip has a buffer zone of 10 m from the plot edges and a width of 3 m. The areas between the tree strips have a width of 22.75 m. In total, the tree strips cover an area of 900 m2 while the agricultural area is 9,100 m2. Growth was conservatively estimated with a uniform annual DBH increment of 1 cm. At the end of the 60-year rotation period, the AFS with its 15 trees contains an above-ground tree volume of over 100 m3, more than 60 t of dry above ground biomass and about 30 t of sequestered carbon in that biomass (Table. 4). The sketch should also be of help getting an impression on the development of the tree crowns over time. While at the beginning the crown diameters are quite small, at the end of the rotation period 37.85% of the area is covered by the trees.

Fig. 5
figure 5

Sketch of an example AFS at age 20, 40 and 60. The light green rectangles represent field strips, the thick green lines represent the tree strips, and the circles represent the single tree crowns. The age is given in years, the diameter at breast height (DBH) and the crown diameter (CD) are given in cm and m, respectively. (Color figure online)

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Table 4 Modelled parameters of an exemplary AFS through time
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Discussion

This study combined TLS and QSMs with the aim to derive total tree volume non-destructively. This combination has been used in numerous studies to estimate above-ground woody volume of single trees. In most cases, QSM-derived data corresponds well with field-based measurements and display only a slight over- or underestimation (Hackenberg et al. 2014; Calders et al. 2015; Gonzalez de Tanago et al. 2018). However, a recent study found this method to potentially lead to an overestimation of volume (Demol et al. 2022). To avoid this, we applied several mitigation techniques proposed by the authors. In addition, we assume that our TLS data is of higher quality, since we surveyed solitary trees in AFS, in contrast to Demol et al. (2022) who faced more difficult conditions since they scanned in forest stands. Due to the unobstructed view caused by the wide spacing between the trees we experienced less occlusion in the point clouds. Likewise, point cloud registration was most likely facilitated by a greater number of adjacent artificial structures that help the algorithm to determine fixed points for linking scans from different positions. Therefore, we expect our volume estimates to be very accurate.

In general, all models except the models on crown base height and the proportion of bole volume showed fair to excellent fits. Furthermore, the residuals of all models were distributed normally and all predictors and predictor interactions showed significant (\(p\) < 0.05) effects on the response variables. These results suggest that appropriate models were selected. The derived models are, however, limited in their applicability. Tree growth is a function of both internal factors, such as provenance, and external factors, such as management and competition as well as environmental conditions. Given the many different systems that can be labeled as AFS, the external factors can vary greatly between individual AFS. For example, crown structure is influenced by tree density and species composition (Pretzsch 2014; Li et al. 2022). Due to this dependence of tree growth on external factors, the models presented here are likely limited in their applicability to similar systems where there is little to no pruning and hardly any competition due to large tree spacing. When applied to dissimilar AFS, model predictions should be considered with caution. Modelling intensively pruned trees, for example, might not be possible with our allometric models. In general, pruned trees are difficult to model as different pruning intensities and types, e.g. whorl-wise or selective pruning, have different effects on tree growth (Springmann et al. 2011). Furthermore, although we considered model behavior in the extrapolation range up to 100 cm DBH during model selection, model uncertainty increases outside the interpolation range. Still, the models provide a solid basis for planning of all AFS which include unmanaged walnut trees without competition.

Tree height & crown

The allometric functions describing single tree height and crown parameters showed excellent fits (Table. 1), except for the model describing CBH (Adj. R2 = 0.45). The latter was expected since CBH is a product of management and other external factors rather than species or dimension. Since the surveyed trees were located at different sites, they were likely managed differently, which in turn resulted in heterogeneous CBHs. None of the trees measured appear to have been pruned with the aim of producing valuable timber, and few of the trees appear to have been pruned at all. In these few cases, apparently selected branches were removed to facilitate management of the agricultural area around the trees. The remaining models are particularly important for designing new agroforestry plots. Crown diameter, for example, may be used for deriving the required planting distance between trees when aiming for a specific DBH. However, the provided crown diameter model can only be used to plan tree spacing in similar systems with unpruned trees where low crown competition is desired. Crown competition can lead to dead branches and thus deteriorate timber quality. For instance, aiming for a DBH of 60 cm for valuable timber production would mean that the crown width of such a tree would be 20 m. As a consequence, a tree spacing of approximately 20 m should be aimed for. Tree height and crown projection area, on the other hand, can be used to assess tree shading, which is a major concern for farmers, because shading of crops might reduce soil temperature which in turn could lead to later seed germination and later ripening of crop fruits. However, shading may also be beneficial in the future due to climate change in order to retain soil moisture and reduce heat stress to crops or, in case of silvopastoral systems, help animals with the regulation of body heat during hot periods. To obtain high resolution insolation predictions, shading models based on 3D data can be used (van der Zande et al. 2010; Oshio and Asawa 2016; Cifuentes et al. 2017; Rosskopf et al. 2017; Bohn Reckziegel et al. 2021).

Tree volume

Gathering volumetric information is necessary when deriving biomass estimates from TLS data. To obtain separate wood and bark volumes, we fitted and applied a simple bark thickness model. The model displayed an excellent fit, although the only predictor variable intentionally was diameter over bark. However, bark thickness is known to vary with tree age and site (Sonmez et al. 2007). This is something we did not investigate. To obtain a more generalizable function, bark thickness should be studied on individuals at different sites with differing environmental conditions and tree dimensions in the future. The regression relating total above-ground volume to DBH also showed a very good fit (Table. 2). Allometric functions for single tree volume are important for deriving biomass, but can also be used to estimate revenue from timber sales at the end of the rotation period. The model describing the relative volume of the branch-free bole compared to the total volume had a much lower pseudo R2 value (Appendix Table. 5). This is likely due to the fact that the model implicitly included CBH, as the volume of the branch-free bole is largely determined by the CBH. As previously mentioned, CBH is heavily impacted by management practices. For example, pruning low branches raises the CBH, increases the length of the branch-free bole, and thus, increases the revenue when the timber is sold (Balandier and Dupraz 1998).

Tree biomass & carbon

Using separate wood and bark volumes, we derived the total above-ground biomass by combining volume with the respective basic density. The basic density of walnut wood, which was calculated from the measured dry density and the volumetric shrinkage, is consistent with values found in literature (Abdulqader et al. 2016; Ostafi et al. 2016). There were no references available on bark density of common walnut. Although the incorporation of bark is important for carbon modelling (Neumann and Lawes 2021), values for bark density for different species are still lacking. Since density is known to vary within and between trees of the same species (Saranpää 2003) and with age (Lundqvist et al. 2018), further measurements of wood and bark density would be valuable for estimating the carbon storage potential more precisely. The resulting allometric functions describing total above-ground dry biomass (Table. 2) and the proportion of bark or wood biomass (Appendix Table. 5) showed excellent fits. In accordance with Hamilton (1975), the share of bark biomass of the total biomass decreased with increasing tree dimension. Although biomass is mainly of interest for deriving carbon and nutrient content, it is also relevant when producing wood for bioenergy, as wood is often sold by weight in the energy sector (West 2004), or for developing production estimates for non-wood forest products.

By combining wood and bark biomass with the respective carbon content from the nutrient analyses, we determined the total carbon content per tree. The relative carbon content of wood and bark differed and varied depending on the diameter of the samples (Table. 3). Our regressions describing total above-ground carbon (Table. 2) and the proportion of carbon stored in wood as opposed to bark (Appendix Table. 5) exhibited very good fits. The total above-ground carbon function can be used to estimate the in-situ carbon storage potential of AFS. A single tree with a DBH of 60 cm, for example, may store up to 2 t of carbon in its above-ground woody parts (in metric tonnes). CO2 equivalents can be easily calculated from carbon content with a factor of 3.67 if aboveground CO2 is a variable of interest, e.g. in endeavors to render agriculture as more carbon neutral by adding trees to crop systems. The proportion of carbon or CO2 stored in wood is particularly relevant when assessing ex-situ carbon storage in durable timber products such as construction or furniture, and can even be improved in terms of carbon substitution potential by a cascading use. However, when applying our carbon function to model carbon in AFS, additional consideration should be given to below-ground biomass and the influence of trees on soil organic carbon. While the below-ground biomass is usually assumed to be about 30% of the above-ground biomass, the impact of trees on soil organic carbon depends on several factors such as soil, climate and management (Abbas et al. 2017).

Tree nutrient content

Based on the biomass and the results of the nutrient analyses, the total nutrient content of five macro nutrients was determined for each tree. As nutrient concentrations in trees are influenced by external factors such as climate (Wu et al. 2007; Sardans and Peñuelas 2013), these results may not be transferable to sites subject to different climatic conditions. Moreover, an analysis of nutrient concentrations in more diameter classes and trees could further increase the accuracy of the estimates. The results of the nutrient analyses showed differences both between bark and wood and between the diameter classes of the samples. The most striking difference was the calcium content of bark, which was up to 30 times higher than that of wood in the same diameter class. It has been previously reported for other deciduous tree species that bark contains significantly more calcium than wood (Wang et al. 1996; Balboa-Murias et al. 2006; Morhart et al. 2016). In accordance with these studies, the bark tissue had an overall higher relative nutrient content compared to the wood. In our case, the magnesium content in the smallest diameter class was higher in wood than in bark. This might be attributable to our small sample size and to the fact that we used composite samples. We found the total nutrient content per tree to follow the order of Ca > N > K > Mg > P. Morhart et al. (2016) reported the same order for wild cherry (Prunus avium) trees sampled in the same region. The derived nutrient-related models showed good fits and can be used to estimate nutrient loss from the system after harvesting trees or when modelling nutrient cycling. The models can be used to calculate the amount of selected nutrients, e.g. a single walnut tree in the investigated AFS with a DBH of 60 cm has a calcium content of around 42 kg in its above-ground woody biomass (Appendix Fig. 7).

Conclusion

Common walnut is an interesting choice for AFS as it provides valuable timber and non-timber products. However, basic information on wood, bark and the structure of walnut trees in AFS is lacking. Therefore, we investigated single tree parameters of walnut trees that are important for AFS management and planning. Based on TLS data and QSMs, we derived several allometric functions. These functions can be employed to plan the establishment of AFS, estimate profit and nutrient loss upon tree harvesting, and assess carbon sequestration potential, among other uses. Although the auxiliary data necessary to derive these functions were obtained from few individuals, the functions provide a basis for decision making and further research. To enable farmers to make informed decisions about which tree species to use, further studies on other tree species are needed. For these future studies, the workflow presented here could serve as reference.