Multi‐trait point pattern reconstruction of plant ecosystems

Plants interact locally in many ways and the processes involved (e.g. competition for resources, natural regeneration, mortality or subsequent succession) are complex. These processes give rise to characteristic spatial patterns that vary over time. The corresponding spatial data, that is the locations of individuals and their specific characteristics (e.g. trees of a certain species and their diameters), are known as point patterns, and their statistical analysis can be used to study the underlying processes and their changes due to environmental scenarios. A special application of point pattern analysis is their numerical reconstruction, which is classically used (a) to generate null models that can be contrasted with observed patterns and (b) to evaluate the information contained in observed data using various summary statistics. Sometimes, the reconstructed datasets are also used to initialise individual‐based or agent‐based plant models with realistic but artificially generated data in order to analyse or forecast the development of plant systems. Previous reconstruction methods of point patterns consider only one mark, or they consider several marks but neglect their correlations. We introduce a method that considers individual locations and two marks simultaneously (in our example information on tree species, and diameter at breast height). The method uses different summary statistics of the second‐order point pattern analysis, such as the pair correlation function and the mark correlation function. By successively modifying the reconstructed spatial pattern, the distance (also called energy), measured in terms of differences in the summary statistics between the generated pattern and the observed pattern, is minimised and a high statistical similarity is achieved. After testing the method on different datasets, the suitability of our method for reconstructing complex spatial forest stands, including the spatial relationships of all considered marks, is shown. The presented method is a powerful tool for generating point pattern data. With minor changes, it even enables the reconstruction of forest stands and plant systems larger than those used to collect the inventory data, and although we used two marks only to demonstrate the power of the method, it is easy to include more marks.


| INTRODUC TI ON
The spatial arrangement of points in a two (or three) dimensional space is an important fingerprint of underlying processes and thus considered valuable information to study the functioning of the systems in which they occur.The distribution of plants in a given area is often a result of their local interactions with the environment and neighbours.For example, processes such as seed dispersal often lead to clustered patterns, while competition for light and nutrients can result in a more uniform distribution.Other processes such as regeneration and mortality can superimpose these effects (e.g. Brown et al., 2011), so that the ultimate goal is to disentangle those processes which explain the patterns best.
For a number of applications, however, the reverse approach, namely the generation of patterns according to predefined summary statistics computed for an observed point pattern can be also useful (Wiegand et al., 2013).This approach, known as pattern reconstruction, generates spatial configurations of points as null models which can be contrasted with observed patterns in order to understand the driving factors and underlying processes.Reconstructing point patterns can lead to faster and better adaptation than point process models because it can easily include many characteristics, while still involving sufficient, albeit less, stochasticity.For example, Arnell et al. (2021) tested spatial associations between life stages for positive or negative associations using null models; Wang et al. (2016) used null models to investigate, among other things, whether locally dominant species have more predictable biotic neighbourhoods; and Jacquemyn et al. (2012) used null models to test whether positive spatial associations exist in different bivariate patterns for different orchid species and their hybrids.This approach, though, still lacks multiple marks and their correlations, and thus does not fully exploit all available information.Spatial point patterns are data that typically consist of the locations of entities (the points) and, optionally, their specific characteristics, which are referred to as marks.Typical examples are the x and y coordinates of trees (the points) and information on species and stem diameter.Further examples are grass or coral communities (locations of organisms specified by species), galaxies in the universe or settlement areas with specific attributes.A range of spatial summary statistics have been developed to characterise point patterns both with and without marks (Cressie, 1993;Diggle, 1983;Ripley, 1981;Stoyan & Stoyan, 1994).They can provide information about more subtle differences in spatial structures that are otherwise unrecognisable.
The description of a forest structure as a marked point pattern requires full surveys of all trunk positions and the corresponding tree attributes.Despite improved techniques in high-resolution aerial photography (Garzon-Lopez et al., 2013;Ullah et al., 2020), unmanned overflights (Horning et al., 2020) or terrestrial laser scanning (Liang et al., 2016), such surveys are still scarce because of the associated costs.This is true, in particular, for larger forest stands and for forest regeneration, that is plants <1.30 m in height.Still, a number of larger inventories are available, including data from the ForestGEO network (Davies et al., 2021), with the Barro Colorado Island (BCI) 50 ha plot (Condit et al., 2019) being the most prominent example, and data obtained by local actors like the Sächsischer Staatsbetrieb Sachsenforst, Germany (cf.Axer et al., 2021).So far, spatially explicit stand data are more often available for smaller sample inventories.Here, the locations of the plants need to be simulated according to their spatial distributions in real forests, and the marks assigned to them must consider the observed correlations (Pommerening & Stoyan, 2008).Since the ultimate goal is the generation of an artificial dataset, which has the same statistical features as the reference (Pretzsch, 1997), the reconstruction itself should consider multiple summary statistics that represent such features in a complementary way.When analysing point patterns, a variety of different summary statistics are available, including first, second and higher order statistics as well as statistics for unmarked and marked patterns.
According to Wiegand et al. (2013) Point pattern analysis is an established field in spatial statistics and has consequently been presented in many books and publications (Baddeley et al., 2016(Baddeley et al., , 2021;;Ben-Said, 2021;Diggle, 1978;Illian et al., 2008;Pommerening et al., 2000;Ripley, 1976Ripley, , 1977;;Stoyan & Stoyan, 1994;Velázquez et al., 2016;Wiegand & Moloney, 2014).Also, the reconstruction of unmarked and marked point patterns has already been addressed by several authors (Bäuerle & Nothdurft, 2011;Hesselbarth, 2021;Illian et al., 2008;Koňasová & Dvořák, 2021;Nothdurft et al., 2010;Pommerening & Stoyan, 2008;Tscheschel & Stoyan, 2006;Wiegand & Moloney, 2014) but correlations between different marks have rarely been considered, and the reconstruction usually involves only single marks or several marks that are independently added to the simulated trees (Hesselbarth, 2021;Pommerening, 2006;Pretzsch, 1997).A direct consideration of the correlations between marks can nevertheless be necessary to achieve a realistic representation of the forest structure.Two examples: (1) the diameter at breast height (dbh) can be age-and species-dependent, but is often also determined by the local constellation of trees.(2) The occurrence of species clusters or characteristic regeneration patterns may be influenced by the constellation of the upper layer of the canopy or the distance to mother trees.Pommerening and Stoyan (2008) have already presented a suitable reconstruction method that takes into account different tree species and their associated dbh distributions.The spatial distribution of points is considered in the form of nearest neighbour summary statistics.This method is suitable for investigating the underlying spatial dependence of points (e.g.clusters or aggregations) but neglects, for example, the statistical relationship between points in the local neighbourhood.For the latter, the pcf would be more appropriate as it measures the probability of finding a point at a given distance from another point.A combination of pcf, K function, D k function and H s function would ensure that different forms of spatial distributions (e.g.hyperdispersion patterns) can be correctly reconstructed (Wiegand et al., 2013).It is also crucial that correlations between marks (the attributes of points) such as dbh and tree species are preserved.This could be achieved, for example by additionally considering a mcf (Pommerening et al., 2000;Wiegand & Moloney, 2014), allowing the reconstruction of patterns in which, for example, the regeneration of one species depends on the occurrence of another species, or the diameter distribution is linked to the species distribution.
The reconstruction method presented in the R package 'shar' (Hesselbarth, 2021) made an important step forward in this direction.It takes both pcf and mcf into account.Different marks, however, are reconstructed separately from each other and added to the points afterwards.This leads, for example, to a loss of the respective species-specific diameter distribution.
A combination of the advantages of the two methods presented by Pommerening and Stoyan (2008) and Hesselbarth (2021) is therefore desirable.To the best of our knowledge, such an approach, in which the reconstruction of a point pattern simultaneously captures several attributes of the points and their spatial correlations, has not yet been investigated.The present study addresses this challenge.
Using forest inventory data as an example, we present a workflow that can be used to (a) construct null model patterns for spatial point pattern analysis and (b) generate artificial datasets suitable for initialising forest ABMs and other stand simulators.In addition, the workflow could help to reduce the time and cost of complete forest inventories of patterns to be reconstructed, as the method can be used to transfer sample plots to larger areas.We first present our methodology and then demonstrate its suitability for reconstructing patterns from field data.

| MATERIAL S AND ME THODS
Our workflow is inspired by the methods implemented in the R package 'shar' (Hesselbarth, 2021), see, in particular, the reference manual of that package, and those described in Pommerening and Stoyan (2008).It takes into account the pcf, mcf and optionally the K, D k and H s functions and currently assumes a homogeneous point pattern.This means that the probability of all point configurations and properties is the same throughout the area.Any spatial variation is due to random effects or, for example, processes such as competition or seed dispersal that follow the same rules everywhere (Stoyan & Stoyan, 1994) and are not influenced by underlying factors (e.g.gradients in water availability).For the reconstruction of inhomogeneous point patterns, see Wiegand et al. (2013) and Koňasová and Dvořák (2021).
In the following, the term diameter refers either to the diameter at breast height (dbh) or to the root neck diameter depending on whether the tree is in the upper or lower canopy layer.In the mark correlations, we distinguish between correlation and local correlation.By correlation we mean the statistical relationship between different points and their marks (i.e.not the classical Pearson correlation coefficient).Local correlation means the statistical relationship between different marks at the same point.The method is implemented in the free statistical programming language R (R Core Team, 2022).All script files, are available in a ZENODO online repository (Wudel, Schlicht, et al., 2022).

| Data used
This study uses datasets of four different locations (see Appen- and is located in the Neusorgefeld forest (compartment 5138) in Brandenburg, Germany.The plot includes 418 trees, with 97 trees in the upper and 321 in the lower canopy layer.For each tree, the species (Pinus sylvestris, Quercus petraea (Matt.)Liebl or Sorbus aucuparia), stem coordinates, trunk height and diameter were recorded.We applied a procedure, described in Appendix D.1, to artificially modify the dataset so that it contains features considered important for a calibration of probabilities of actions (Wudel, Huth, et al., 2022).The three other datasets were used to assess the suitability of the method.Dataset 2 was provided by the Chair of Forest Growth and Production of Wood Biomass at the TU Dresden (Fichtner & van der Maaten-Theunissen, 2022)

| Reconstruction of point patterns taking into account local correlations among marks
The following example illustrates the initialisation and subsequent processes required for the reconstruction of point patterns.We assume a reference pattern consisting of five points, characterised by x-and y-coordinates [m], dbh [mm] and species.This small number is for expository purposes, and meaningful results are usually only expected for patterns containing larger numbers of points.
The observation window has a dimension of 10 × 10 m (see Appendix A.1).The reconstruction starts by generating an initial pattern containing the same number of points as observed in the reference pattern.The coordinates of each point are drawn from a uniform distribution.Each of the marks, which can be numerical (e.g.dbh) or nominal (e.g.species), is selected independently from the corresponding empirical distribution in the reference pattern.Internally, the procedure transforms nominal marks into a set of indicator variables, so that all marks that enter the computation are effectively numerical.Next, the summary statistics are determined for both the reference pattern and the initial pattern.By default, these are second-order spatial statistics related to the pcf and the mcf (Wiegand & Moloney, 2014).This computation is fully described in Appendix C and is optimised for efficiency.The full R code is found in the ZENODO online repository (Wudel, Schlicht, et al., 2022).
Optionally, the following summary statistics based on functions Kest, nndist and Hest, respectively, from the R package 'spatstat' can be included in the reconstruction: a.The K function, which is the cumulative counterpart of the pcf and examines the spatial structural composition of the point patterns.It allows to assess whether the point patterns are clustered or not (Stoyan & Stoyan, 1994).c.The H s function represents the distribution function of distances from an arbitrary test location to the nearest point of the pattern (Illian et al., 2008).The H s function is useful for characterizing point patterns with larger areas without points, as it essentially measures the distribution of the size of the gaps in the pattern (Wiegand et al., 2013).
The higher efficiency in the computation of the default secondorder statistics is achieved by updating only the contribution of modified points to the various statistics in each step.These are computed within the relevant distance determined by r .The distance r can be thought of as the radius of the area in which a tree has a direct influence on its neighbours.For example, this could be the crown radius or the radius of the root ball, but could also be larger to properly represent longer-range dependencies.See Appendix B for details.Table 1 summarises the specific parameters and their default values of the described functions.
The initial energy, which measures the distance (difference) of the reconstructed pattern from the reference pattern, is now calculated by default as a weighted sum of L 1 distances, that is mean absolute the differences of the summary statistics described above, and corresponding differences of the local correlations.The L 1 distance can be changed to, for example to a squared L 2 distance (sum of squared differences) or to other pth powers of L p distances.In subsequent steps, the energy function is used as an evaluation criterion to decide whether the new pattern is better than the previous one with respect to the reference pattern.It is possible to weight the different statistical measures in the calculation of the energy, and this makes it possible to give special consideration to certain aspects in the reconstruction.For example, a higher weight for a particular pair of marks means that the correlation considered by these functions is considered more important than other features of the point pattern, and so, for example, the mcf between tree species and diameter is considered more important than the spatial distribution of points considered by the pcf.
By default, all weights are set to 1 so that the summary statistics have equal weights.The K function, D k function and H s function can be given different weights when activated.The weights of the default second-order statistics can also be set when the function is called.
Particular combinations of features can be weighted differently, for example the correlations where the metric mark is included, such as diameter and tree species of two points, or the correlations of diameter and diameter of two points.The same is possible when considering a point with itself, which corresponds to the local correlation.
The full list of weights for all combinations of marks can be found in Table 2.
The aim is now to gradually minimise the energy value by successively modifying the pattern by six types of actions, one of which is chosen with certain probabilities p x , x = 1, … , 6, in each step of the simulation.Table 3 lists the possible actions for energy minimisation.
The actions in detail are as follows: a. move_coordinate: The x and y coordinates of a randomly chosen point are changed.New coordinates are generated from a twodimensional normal distribution with an expected value given by the current coordinates and a standard deviation defined as the width and height, respectively, of the observation window divided by the number of simulation steps already completed.This standard deviation was chosen so that larger deviations are possible at the beginning of the simulation but become smaller afterwards.The underlying assumption is that as the simulation progresses, the reconstructed pattern becomes successively better adapted to the reference pattern, so a larger shift of the coordinates at the end would not lead to a better adjustment.This action tends to manifest itself as a kind of 'wobbling' with increasing simulation time.c(1, 1, 1, 1, 1, 1,  1, 1) w_statistics

[-]
A vector of named weights for optional spatial statistics from the 'spatstat' package to be included in the energy calculation.This may include Dk, K, Hs. c() TA B L E 1 Parameters of reconstruction procedure, with units, description and default values.They are implemented in the R-scripts accessible via the ZENODO online repository (Wudel, Schlicht, et al., 2022).
f. pick_mark_two: The nominally scaled mark (species) of a random point is replaced by a value drawn randomly from its distribution in the reference pattern.
After an action has been carried out, the energy is calculated to check if the change led to an improvement.Only in this case, the new pattern is retained; otherwise, the change is discarded and the preceding pattern is kept.The whole procedure is repeated until the specified number of simulation steps has been completed (see Figure 1).

| Validation of the reconstruction method
The validation was carried out using the datasets described in Section 2.1, as follows.

| Example of a calibration
Dataset 1 was used to illustrate a possible calibration of the action probabilities and the weighting of the statistical measures of the reconstruction procedure for a defined dataset.The procedure is described in Appendix D.1.The results of the weighting and an example reconstruction with the corresponding calibration are given in Appendix D.2.The results of the calibration of the probabilities can be found in Appendix D.3.

| Reconstruction method test procedure
Datasets 2-4 were used for this purpose.These were reconstructed as far as possible using the standard settings of the reconstruction method (see Section 2.2).This means that the summary statistics described in Section 2.2 and Appendix C were used for the reconstruction without the optional Kest, nndist or Hest functions from the R package 'spatstat'.For all three datasets, the number of simulations to be performed was increased to 100 (n_ repetitions, default setting 1) in order to achieve a certain level of statistical significance.In addition, rmax was increased from 5 to 25 m, assuming that this corresponds to the maximum area of influence of a tree in the datasets.The maximum number of simulation steps max_steps was increased to 100,000 for datasets 2 and 3, and to 600,000 for dataset 4, to ensure that on average each object is affected by at least 10 potential modifications during the reconstruction.The reconstruction results are then compared to the reference pattern and the summary statistics are compared, see Section 3.

| Simulated patterns
While we believe our datasets are sufficiently close to homogeneous point patterns as required by our methods, we cannot strictly exclude potential inhomogeneities in the real patterns that might affect the ability of the summary statistics to characterise the spatial structure of the point patterns.To exclude such effects, additional simulated patterns that are homogeneous by design (see Wiegand et al., 2013) were reconstructed as well.The description of these patterns and the results of the reconstructions can be found in the Appendix E.

| RE SULTS
First, we present the results of the 100 reconstructions of dataset 2. The following Figure 2  and pcf (d), even over the range considered in the reconstruction (25 m).For the mcf of the species (e) and the diameter mcf (f) there are smaller systematic deviations (about 2%).This is probably due to the fact that the weighting is not optimal with regard to the diameters, while the number of steps is still quite small.Specifically, there are many small diameters (0.005-0.1 m) in this sample, which is much smaller than the species indicator variables (value 1 or 0) and so contributes less to the energy.Thus, the reconstruction perfectly reproduces the numbers of species but shows deficiencies, in particular, with regard to the large diameters of the Cecropia insignis trees.On average, a reconstruction of this dataset with descriptive parameters took 67:14 min.on the same machine as before.
F I G U R E 1 Flow chart of point pattern reconstruction with two marks (metric and nominally scaled).The actions to minimise the energy (make the simulated patterns statistically more similar to the reference patterns) are carried out with different probabilities (p x , x = 1, … , 6).

| DISCUSS ION
The aim of this study was to develop a method for reconstructing marked point patterns that simultaneously considers multiple marks and correlations between them.Our method has the fol- Europe.This is of course species dependent and may be incorrect in other geographical latitudes and environmental settings.Also, it does not capture effects that occur on a larger scale such as seed dispersal.Furthermore, while this value is assumed to be the same for all trees, it would ideally be determined for each tree depending on its individual characteristics (height, crown diameter or root ball diameter).In the present study, this was not considered in order to keep the computation time and the complexity of the procedure as low as possible.In addition, the chosen weights might need to be adjusted for other forest stands, but they provide very good results for the examples shown here.
With minor changes in the R code, the reconstruction method is regeneration are practically non-existent, but increasingly important for forest growth models, such an extension is of high potential value.
In conclusion, the method presented is suitable for generating statistically plausible variants of empirical forest stands which can be used, for example, for the initialisation of individual-based forest growth models.While we focused on forest stands, our method is also suitable for reconstructing other point patterns in which two or possibly more marks are significant.In principle, any spatial system of random objects having fixed positions and two or possibly more marks is a candidate for such a pattern.These could be, for example, grass or coral communities but also settlement areas.
Methods are thus needed to reconstruct complete forest stands based on limited sample information, or to generate artificial stands with patterns statistically similar to those in the observed data.Indeed, reconstructed marked point patterns can be used for several purposes, (a) to construct null model patterns for spatial point pattern analysis, (b) to initialise computer-based forest simulators like agent-based models (ABMs) with realistic forest data and to evaluate their suitability or (c) to represent predicted forest structures for the environmental and management scenarios of interest.
, promising candidates for the reconstruction of point patterns are the intensity function, the pair correlation function (pcf), the expected number of points within a distance r from the typical point (K function), the nearest neighbour function (D k function) and the spherical contact distribution (H s function) In addition, there are mark correlation functions (mcf) for the observed of the marks.
dix A.1 Visualizing the locations of the surveyed areas of datasets 1-3).The first dataset (dataset 1) is based on data obtained at a 50 × 50 m plot that belongs to the study sites of our project Effective spatial modelling and simulation of silviculturally controlled mixed regeneration under changing climate conditions (VERMOS) . The data was obtained from 308 trees at a 70 m × 140 m marteloscope area in the Naundorf district (compartment 710 b1) located in the forest Tharandter Wald (Saxony, Germany).Dataset 2 and dataset 3 contain dbh [in mm and cm respectively], tree height [m], x and y coordinates [m], and species.The one, referred to as dataset 3, was obtained by the Nordwestdeutsche Forstliche Versuchsanstalt (Spellmann, 2022) and contains data from a 160 m × 160 m forest plot located in the Reinhausen forest (compartment 3002j, southern Lower Saxony, Germany).It includes all trees with dbh (including bark) ≥ 70 mm.Dataset 4 is a subset of the data form the seasonally humid tropical forest on BCI in Panama (Condit et al., 2019).A 50 ha plot was established in 1982 and has been continuously monitored.All trees >1 cm diameter at breast height (dbh) have been mapped, marked and measured every 5 years since 1985.Following Wiegand et al. (2013) we used eight species from the 2010 BCI censuses, which show different patterns (from clustered to hyperdispersed).The dataset thus consists of 60,141 points over an area of 50 ha.
b.The D k function, which is the cumulative distribution function of the distances to the kth neighbour.It indicates the probability that a typical point of the pattern has k neighbours within a given distance r(Illian et al., 2008).
b. switch_coords: Coordinates of two randomly chosen points are swapped.c. exchange_mark_one: Marks measured on the continuous scale (i.e. ) of two randomly drawn points of the point pattern are exchanged.d. exchange_mark_two: The nominally scaled marks (i.e.tree species) of two randomly drawn points are exchanged.e. pick_mark_one: A new value for the mark measured on the continuous scale ( ) is generated based on the smoothed empirical distribution of the reference pattern.
compares the reference pattern (a) with a reconstructed pattern (b).It is easy to see that the pattern characteristics have been well reproduced.For example, it is clear that Picea abies occurs at high intensity only in limited areas of the reference pattern and that this has been reproduced.Similarly Quercus TA B L E 2 List of the weights in the energy function.Possible actions to minimise energy (the measure of statistical differences between generated and reference pattern) and a brief description of the changes they trigger in the generated point pattern.Metric mark of a point is recreatedp total =100 % petraea occurs mainly at the edge of the reference pattern, and this feature has also been reproduced well.In addition the figure includes various summary statistics (c)-(f).These also show that the reconstructions (grey line = reconstructions, black solid line = reference) work very well.In the range up to 25 m considered in the reconstruction, the deviations from the reference in the K function (c) and pcf (d) are negligible.The mcf of the species (e) and the diameter (f) show only small deviations, but these become much larger once the range up to 25 m is exceeded, illustrating the effectiveness and influence of the mcf's in the reconstruction method.A weighting of these functions could lead to even better fits.On average, a reconstruction of this dataset with the given parameter settings took 2:47 min on a standard standard office notebook (AMD Ryzen 7 PRO 4750U preocessor, 1700 MHz, 8 cores, 16 logical processors with 32.0 GB RAM).The results of dataset 3 are shown in Figure 3.It can be seen that certain clusters of species of the reference pattern can be reproduced.The summary statistics clearly show the high quality of the reconstruction.In the range up to 25 m considered in the reconstruction, there are no deviations from the reference in the K function (c) and pcf (d).Marginal deviations can be seen in the mcf of the diameter (e), while the mcf of the species (f) is also almost identical to the reference.The mcf's do not deviate significantly from the reference pattern as long as the range of 25 m is not exceeded.On average, the reconstruction of this dataset using the given parameters took 7:55 min on the machine mentioned earlier.Finally, the results from dataset 4 are presented (see Figure 4).A visual evaluation of the reference (a) and the reconstructed pattern (b) is more challenging because of the very high number of trees (60141).The summary statistics, however, allow a clear evaluation of the reconstruction.The reconstructions show no relevant deviations from the reference pattern in the K function (c) Figure E.3.7 in Appendix E, which shows the K function, D k function and H s function for a reconstruction not involving those functions,and the effect when they are included in the reconstruction.However, to confirm this conclusively, the influence of the mcf would need to be further investigated.It could be confirmed that it is essential to consider the correlations of the marks in the reconstruction at the same time in order to achieve the statistical properties of flexible enough to reconstruct tree patterns in areas larger than the original forest stand, although, depending on the size or shape, edge corrections might become necessary.It is also possible to consider more than two marks.In addition, it should be possible to reconstruct regeneration trees observed only locally in sample inventories under overstorey of the remaining area, which can be measured by laser scanning.As complete inventories of forest stands with plant F I G U R E 4 Results of the 100 reconstructions of dataset 4; (a) reference pattern (b) an example of a reconstruction based on default second-order statistics (see Appendix C for details).Summary statistics (c)-(f) were generated using the R package 'spatstat': (c) K function; (d) pcf; (e) mcf of the species, where the test function has value 1 if both species coincide and 0 if not; (f) mcf of the diameter, the test function being the usual product.For distances r up to r max = 25 m (indicated by the vertical dashed line in (c)-(f)).The black solid lines represent the reference curve and the grey solid lines represent the 100 reconstructions.The dashed black line is the mean of the 100 reconstructions.