Jianjun Li·Kaiwen Zhu·Shuai Liu·Dan Li·Gui Zhang·Xing Liu·Weiming Yang

Abstract It is important to quantify and analyze forest spatial patterns for studying biological characteristics,population interaction and the relationship between the population and environment.In this study,the forest spatial structure unit was generated based on the Delaunay triangulation model (DTM),and the weights were generated using the comprehensive values of the tree diameter at breast height,total height and crown width.The distance between neighbors determined by the DTM was weighted to transform the original coordinates of trees into logical coordinates.Then,a weighted spatial pattern (WSP) was developed.After weighting,the neighboring trees were replaced,the replacement ratio was 38.3%,and there was 57.4% of the central tree.Correlation analysis showed that the uniform angle index of the WSP was significantly correlated with the tree size standard deviation under uniformity (r =0.932)and randomness (r =0.711).The DTM method not only considers the spatial distance between trees,but also considers the non-spatial attributes of trees.By changing the spatial topological relation between trees,this method further improves the spatial structure measurement of forest.

Keywords Delaunay triangulation network model ·Weighting·Tree spatial pattern·Tree size standard deviation

Introduction

Forests are the largest terrestrial ecosystems on earth and an indispensable part of services such as biodiversity maintenance,water and soil conservation,and climate regulation(Sakkatat 2007).Comprehensively and accurately analyzing and evaluating forest spatial structure is important in modern forest resources management.Tree size,species mingling and spatial distribution patterns are the three most important characteristics of forest spatial structure (Goff and Zedler 1968;Hui 1999).Spatial distribution patterns directly ref elct the way that individual trees assemble or scatter in a forest,which may in turn be associated with conditions of competition and utilization of environmental resources among neighboring trees (NT).These patterns not only provide information about tree growth,development,competition and death but also affect,for example,habitat availability,species and genetic diversity,and fire and disease outbreaks(Schneider et al.2016;Carrer et al.2018).However,these patterns are affected by wind,water flow,light intensity,animals and species characteristics,such as type of regeneration and seed dispersal.Besides,other unknown factors,which result from technical limitations of the aforementioned factors,also have inf ulence on them (Gratzer et al.2004;Myster 2012;Parker and Sherwood 2 013;Duman et al.2016).Thus,there have been great challenges regarding improving the reliability of stand spatial pattern analysis and evaluation,particularly for natural mixed forest because of the substantial complex features in forest spatial structure.

A wide range of methods to study stand spatial patterns were proposed over recent decades (Girdler and Radtke 2006;Groen et al.2008;ErfaniFard et al.2008;Thompson et al.2013).These methods can largely be divided into three categories:distance methods,point pattern analysis,and spatial structure unit (SSU) methods.Distance methods include the nearest neighbor index and T-square sampling.Researchers only need to measure the distance between the reference tree (RT) and the nearest tree.Distance methods have been widely applied by researchers because of their simple statistical process and calculation (Clark and Evans 1954;Diggle et al.1976;Girdler and Radtke 2006;ErfaniFard et al.2008;Groen et al.2008).However,their application is still rather limited in stands with a complex spatial structure because only the nearest tree to the RT is considered (Kint et al.2000).Point pattern analysis,also called second-order statistics,is based on the distribution of distances of pairs of points,such as Ripley’s K-function,the pair-correlation function and O-ring statistic,which have been widely applied for analyzing the different-scale spatial patterns of stands over the last several years (Thompson et al.2013;Gu et al.2017).The significance test is typically evaluated by comparing the observed data with Monte Carlo envelopes from the analysis of multiple simulations of a null model (Idohou et al.2016).Point pattern analysis provide more information on the scale of the pattern than statistics that use nearest neighbor distances only (Zhang et al.2015a,b;Fibich et al.2016;Janík et al.2016;Shin et al.2017;Wu et al.2017).However,these point pattern analysis methods involve the technical decision of actual ring width,so their calculation process and edge correction are complicated(Wiegand and Moloney 2004).

In recent years,SSU methods have received much attention,and have been widely adopted and applied in stand spatial pattern research (Wehenkel et al.2015;Palaghianu 2016;Ghalandarayeshi et al.2017;Zhang et al.2018a,b).The premise of these methods for analyzing the spatial distribution pattern is to select neighbor trees that are closely related to the RT.For one of SSU,the UAI describes the degree of scatter for the four nearest neighbors to the RT.Its distribution pattern can be evaluated by comparing the angle between any two neighboring trees and the RT with a standard angle (72°).The UAI has been commonly used in the evaluation of spatial patterns of individual trees and entire stands (Hui 1999).Meanwhile,the coefficient of variation of the Voronoi polygon has been applied in forestry since the rapid development of geographic information technology(Zhao and Li 2017).The Voronoi diagram determines the adjacency relationship between neighbor trees and provides a basis for SSU methods based on neighbor relationship.The Voronoi diagram takes the position of trees as the center and divides the bi-dimensional space according to the nearest principle (Duyckaerts et al.1994).As a result,a Voronoi polygon of a tree includes all the points closer to the specific tree than to any other tree,and two trees are considered to be neighbors if their associated Voronoi polygons share an edge.Maximally equilateral non-overlapping triangles,that is,the DTM,are formed by connecting the trees on both sides of the common edge,that is,the spatial structure unit of each tree is determined.Recently,Voronoi polygons have been used to represent the“influence area”of a tree and the DTM to describe the neighborhood of the RTs (Zenner and Hibbs 2000;Zhao and Li 2017),which guarantees the maximum correlation between the RT and its NTs.Once neighborhoods are thus delineated,not only spatial patterns but also virtually any metric of structural complexity that assesses size differentiation,such as mingling and the competition index,can be quantified at the neighborhood level(Zenner and Peck 2018).

However,most of these studies on SSU only consider the influence of spatial distance on SSU formation,and ignore the influence of trees’ own attributes (DBH,tree height,canopy width,etc.) on SSU formation.A few studies have explored the influence of tree characteristics (DBH,height,canopy width,etc.) on forest spatial pattern,but only limited to some changes of influence area (Voronoi polygon)(Aakala et al.2013;Krejza et al.2015;Li et al.2015;Zhang et al.2015a,b;Cao et al.2016).

In view of the def iciencies of the current spatial pattern analysis,such as the over-dependence on the spatial point position,a weighted Delaunay triangulation method was proposed in this paper to give weight to the spatial point position of trees,including DBH,tree height,crown width and other attributes.Thus,a more detailed forest spatial information description mechanism was established.The objectives of this study were:(1) to generate weighted logical positions on the SSUs determined by the DTM;(2) to test the correlation between the weighted spatial pattern (WSP)and size standard deviation;and,(3) to quantify and analyze the spatial pattern change of an entire stand in Longhushan Park,Chishan Island,South Dongting Lake,China.

Material and methods

Approach based on the Delaunay triangulation weighting method

Triangulation and the SSU

A triangulation is a network of planar control graphs that consists of a series of continuous triangles,which reflect the relationship between discrete points in a plane(Fig.1).The relationship between the discrete points inthe plane formed by the triangulation is used to determine the SSU of the forest:the line between the discrete points indicates the interaction between two points,and a discrete point is connected with a plurality of surrounding points,wherein the center is discrete.The point represents the target tree,and the surrounding discrete points connected to this represent trees adjacent to the target tree(Fig.1).The SSU determined by this method is more flexible than the fixed radius circle method and "1+4"center tree-neighbor tree method (Zenner and Hibbs 2000).

Fig.1 Weighted principle of the triangulation model

Logical coordinates based on weighted triangulation

There are intrinsic relationships between the DBH,tree height and crown width,and the influence degree of these three factors of the competitiveness of trees is different.Cao et al.(2016) found that the weights of the DBH,tree height,and crown width on the competitiveness of trees were 0.38,0.29,and 0.33,respectively,which were obtained using grey relativity analysis.However,these weights vary from stand to stand,species to species.In this study,we used Cao’s method to calculate the weights of the DBH,tree height and crown width of single tree,and then calculated the attribute weight (wi) of each tree in the plot.Typically,the larger the DBH,total tree height and crown width,the larger the corresponding attribute weight and the stronger the competitiveness of the tree in the stand.wiis given by

whereed,eh,andecare the influence weights of the DBH,tree height,and crown width,respectively.Di,HiandCiare the DBH,tree height and crown width of the i-th tree;andandare the average DBH,tree height and crown width of all trees in the sample plot,respectively.

The attribute weight of single tree

Each tree in the forest needs to compete with neighboring trees (neighboring trees below is abbreviated as NTs) to obtain necessary resources and growth space.The interaction intensity between NTs and central tree (central tree below is abbreviated as RT) is related to the size of both NTs and central tree.Therefore,the average value of the attribute weight of center tree (wi) and NT (wj) was considered as the comprehensive weight (wij) between them.If the distance between the central tree and NT is fixed,then the larger the value ofwij,that is,the larger the DBH,tree height and crown width,the more intense the competition between the two trees,the stronger the interaction,and the smaller the relative distance;and conversely,the weaker the interaction relationship between the two trees and the larger the relative distance.

The comprehensive weightwijbetween the center tree i and its neighboring tree j is given by

wherewiis the attribute weight of the center tree andwjis the attribute weight of the neighboring tree.

The movement of the center tree relative to the neighbor tree of the kth tree,and it is calledwhich is given by

There are k NTs around the i-th RT (Same as center treewiabove) in the i-th SSU determined by the DTM,which arej1,j2,j3,…,jk.Starting from the RT,the vectors formed by the locations of the NTs areThe specific weighting method is divided into two scenarios to transform forest attributes into spatial factors:if the value ofwijbetween the central tree and NT is greater than one,this indicates that the interaction between them is strong and the relative distance between them is smaller than the actual distance.The larger the value ofwij,the smaller the relative distance between the two trees.Hence,the RT should move in the same direction as,and the moving distance is expressed by the modulus of.When the value ofwijis less than one,this indicates that the interaction is weak between the central tree and NT,and the relative distance between them is larger than the actual distance.The smaller the value ofwij,the larger the relative distance.The RT should move away from the NT in the opposite direction while the distance is also the size of a module of(Fig.1).The RT has corresponding moving vectors in the same direction or opposite direction relative to its NTs.The sum of all moving vectors of the i-th RT yields vector,which is the total vector of the RT’s movement.The endpoint ofis the logical position of the i-th RT.The distance between trees with a strong interaction relationship decreases,whereas the distance between trees with a weak interaction relationship increases after the corresponding logical position of the central tree is generated.As a result,the UUSs of the trees and their NTs were changed and replaced (Fig.2).

Figure 1 shows the principle of weighting the DTM.Figure 1 a is a plane triangulation model formed by tree position points;the interior of the red box is research area,the exterior is the buffer area;the grey straight line is the edge of the triangular network.Figure 1 b is the SSU marked on the left-hand side;the red solid circle and the black solid circle represent the RT and NT,respectively;the number represents the comprehensive weight between the RT and NT;the blue arrows and green arrows represent the opposite and same direction moving vectors,respectively;the black arrow represents the sum of all moving vectors of the RT,and the red dotted circle represents the logical position of the RT after weighting.

Figure 2 shows the changes of tree neighbors before and after weighting.Figure 2 a The distribution of trees before weighting:the red circle represents the RT,green circles represent the NT of the RT,blue circles represent the NT of the NT and black numbers denote the number of trees;and Fig.2 b The distribution of trees after weighting,the position of the weighted tree changed and the NT of the RT also changed.

Fig.2 Changes of tree neighbors before and after weighting

Spatial pattern analysis

To test the forest spatial distribution pattern and its changes based on triangulation weighted by forests,in this study,the UAI and bivariate pair correlation function were used to quantify the spatial distribution pattern of the stand before and after weighting.The UAI was used to analyze the distribution of trees by comparing the angle formed between any two neighbor trees with the standard angle (T0=72°) (Hui 1999).It is defined as the proportion of the number of angles that are less than the standard angle to the number of the nearest NTs,n is the total number of trees in the sample plot,each tree has four angles with its nearest NTs andZijis the counter variable,which equals one if thejth angle is less than T0and zero otherwise.The larger the value of UAI (Ghalandarayeshi et al.2017),the more clustered spatial pattern of the stand.When the mean UAI of all trees is in the range [0.475,0.517],which shows a random distribution;otherwise,the distribution pattern is classified as clumped (meanUAI>0.517) or regular(meanUAI<0.475):

The pair-correlation functiong(r) is derived from Ripley’s K-functionK(r).The circle ofK(r) is replaced by a ring.The ratio of the forest point density in the ring withr as the radius to the expected density at random states is counted.The advantage ofg(r) is that it can directly explain the local neighborhood density and describe the dynamic changes of the spatial pattern in different scales.It can eliminate the scale cumulative effect of Ripley’s K-function and is more intuitive than the cumulative measurement.The independent expression ofg(r) is defined as the ratio of the probability of the existence of points in any two inf initely small discs with distance r to the probability under completely random conditions (Stoyan and Penttinen 2000):

wheredxanddyare the areas of two small discs;λxandλyare the density of trees in the two discs,respectively;and λ is the expected density of sample plots under a completely random state.For random populationg(r)=1,the clustered populationg(r) >1 and the regularly spaced populationg(r) <1.In this study,the Monte Carlo method is used to perform a simulation of 100 times to obtain a 95% conf idence interval for the upper and lower envelope traces.Theg(r) curve above the upper envelope trace indicates that the spatial pattern of the sample is a clustered distribution,which is randomly distributed between the envelope traces and uniformly distributed below the lower envelope trace.

Competition index

In this study,the competition index was used to analyze the intensity of competition between neighbor trees.The formula is:

whereidenotes the competing trees that represent the central tree,DiandDjare the diameter at breast height (DBH)of the central tree and the competing tree,respectively,andDistijis the distance from the competition tree to the central tree.

Correlation analysis

In this study,the Spearman correlation coefficient was used to test the relationship between the weighted forest spatial distribution pattern and the forest size difference.All analyses were performed using the SPSS 21.0 software package(SPSS Inc.,Chicago,USA) for Windows.If the p-value is less than 0.05,this indicated that there is a significant correlation between the two random variables in the 95%conf idence interval.In this study,the correlation between the standard size difference of forest trees and the angular distribution of weighted post-forest spatial distribution patterns was analyzed under three distribution patterns:uniform,random and aggregate.

Data

Simulated plot data

The forest data in this study were divided into simulated plot data and natural plot data.15 sets of forest sequences were generated using MATLAB and numbered 1—15.There were 400 trees in each group.Diff erent forest tree weights were used to represent the different sizes of trees,and the standard size difference between trees in the group increased as the group number increased:the minimum standard size difference of group 1 was 0.019,and for group 15,the standard deviation of the size of the forest was the largest at 0.565.The 15 groups of forest sequences were combined with uniform,random and aggregate distribution patterns to generate 45 simulated plots with a plot area of 100 m × 100 m Fig.3 a—c.After all the plots were weighted,the weighted spatial distribution pattern was quantified using UAI.Additionally,one control group was set up:the number was 16 and the number of forest trees was still 400.The attribute weight of all trees was the constant value one,and the standard deviation of the forest size was zero.The three distribution patterns of the forest sequence did not need to be weighted.

Fig.3 Three spatial distribution patterns:uniform,random and aggregated

Natural plots

The natural plot data were obtained from a field survey of Longhushan Forest Park in Chishan Island,Hunan Province,China (1 12°18′15′′−1 12°56′15′′E,2 8°113′30′′−2 9°3′45′′N),where the main dominant tree species in the study area wereSchima superb(S.superb),Liriodendron chinense(L.chinense) andCryptomeria fortune(C.fortune).A standard plot with an area of 130 m × 130 m was set up in the study area.It was subdivided into quadrats of 10 m × 10 m.Then the position,species information,DBH,tree height and crown width of all standing trees with DBH greater than 5 cm in these small grids were investigated.Taking the southwestern corner of the plot as the origin of the coordinate axis,the spatial coordinates of trees in the sample plot were recorded as (x,y),where x and y represented the position in the east—west and north—south direction,respectively.The DBH was measured using a ruler to the nearest 0.01 cm.Tree height was measured using an ultrasonic altimeter to the nearest 0.1 m.The crown width was measured with a tape measure to the nearest 0.1 m and the spatial position of each tree was recorded.Additionally,site-related factors such as elevation,slope direction and slope gradient were also investigated (Table 1).

The formation of the forest triangulation model is essentially a process of determining the central tree-neighbor trees together to form the SSU.Because of the influence of plot division,the neighbor tree of the forest part of the plot may be outside the plot;only the triangles formed by all the forests in the plot will have a severe deformation of the edge triangle shape.Additionally,the forest space determined on this basis SSU will also be inaccurate and the statistical results will be inaccurate.Moreover,the smaller the plot area,the smaller the number of trees and the greater the error caused by this edge effect.Therefore,it was necessary to set a buffer of a certain width at the edge of the sample (see Fig.1).The wider the buffer setting,the smaller the error,but the core area was too small,which made the statistics of the forest lose meaning.According to the difference between the simulated sample plot and the actual plot size,the buffer width was set to 10 m and 15 m,respectively.The trees in the buffer were only used as neighbor trees,and the triangular mesh in the buffer was not included in the statistical range.

Results

Correlation analysis of weighted spatial distribution patterns

As shown in Table 2,the UAI values of control group No.16 in the uniform,random,and aggregate spatial distribution patterns were 0.121,0.512,and 0.523,respectively,which represent the UAI values of the original three typical distribution patterns.Fifteen plots of 15 tree size standard deviations that corresponded to the original uniform,random and aggregated distribution patterns were weighted,and the angular dimension values of the stand were different (see Table 2).Under the three distribution patterns,the original stand pattern for uniform distribution,the correlation between tree size standard deviation and the weighted post stand UAI was the strongest (r=0.892,p=0.000).However,when the original spatial pattern were random and aggregated,the correlation between the standard size difference of forest trees and the UAI values of the weighted forest stands were not obvious (r=0.275,p=0.420 andr=0.408,p=0.111) (see Fig.4).

Fig.4 Correlation between the spatial distribution pattern of weighted stands and tree size standard deviation under three distribution patterns

Table 1 Forest information in actual plots

Table 2 Values of the UAI under three spatial distribution patterns:15 tree size standard deviations after weighting

Change in tree position

Table 3 shows that the average radius of the SSU formed by the actual sample was 3.98 m.The weight of the buffer tree changed greatly:the maximum distance was 67.12 m,the average moving distance was 2.49 m and the proportion of the moving range of the buffer forest exceeded the radius of the SSU by 13.4%.The movement range of forest trees in the core area was small,and the proportion that exceeded the average radius of the forest space structure only accounted for 5.4%.

Weighted forest topological relationship and competition

Table 4 shows that the average number of neighbors in the SSU before and after weighting was 6.00 and 6.07,respectively.During the weighting process,some neighbors were replaced by NTs with stronger interactions.The average number of neighbor trees replaced was 2.38,which accounted for 38.2% of all neighbors in their corresponding SSUs.The neighboring forests with stronger relationships became the new neighbor tree regenerating the SSU,and the average number of neighbors was 2.41,which accounted for 38.9% of all neighbors in the SSU.

The forest competition index change was represented by the difference between the weighted forest competition index and the weighted pre-forest competition index,as shown in Table 5.The difference between the partitions was (− 1.5,− 1],(− 1,− 0.5],(− 0.5,0],(0,0.5],(0.5,1],(1,1.5],(1.5,2].In the ranges (1.5,2],(0,0.5),the number of trees was the most,that is 389.When the difference was greater than zero,the weighting was such that part of the neighbor tree of the center tree was replaced,and the new neighbor tree was centered on the tree.According to the statistics,the forest trees with an increased weighted competition index accounted for 55.3% of the total forests.

Table 6 shows that the forests in the actual plot were divided into three grades according to the diameter of the breast diameter,which were 5—10 cm (greater than 5 cm and less than or equal to 10 cm),10—15 cm,and >15 cm,respectively.The corresponding number of forest trees was 500,115,and 118,respectively.The larger the forest level,the smaller the competition index before and after weighting.The average competition index of different diameter forests increased after weighting,with growth rates of 2.4%,14.2%,and 13.0%,respectively.

Table 3 Tree movement distance after weighting

Table 4 Number changes of neighbor trees before and after weighting

Table 5 Change of the competition index

Table 6 Change of the competition index for different diameter grades of trees

Analysis of the spatial pattern before and after weighting

As shown in Fig.5,all the trees in the actual plot generated corresponding logical positions based on triangulation,and the logical positions of all the trees did not significantly exceed the boundary of the plot;instead,the local forests tended to be close together.

Fig.5 Unweighted and weighted spatial distributions of the plot

Figure 5 a,b represent unweighted and weighted forest distribution patterns,respectively;different size circles represent different sizes of trees;orange,green and blue represent woody,eucalyptus and cedar trees,respectively;and gray lines indicate the association of the central tree formed by triangulation Fig.6 shows that the UAI and double correlation function were consistent with the test results for the spatial distribution pattern of forest trees in the core area of the actual plot.Before weighting (Fig.6 a),the UAI value was 0.509,which is a random distribution in the range of[0.475,0.517];the double correlation functiong(r) curve was located between the upper and lower envelope lines and was randomly distributed.After weighting (Fig.6 b),the UAI value was 0.544,which is greater than 0.517,which is the aggregate distribution;the double correlation functiong(r)curve was clearly higher than the upper envelope trace on the small scale of 0—5 m,and appeared to be strong.

Fig.6 Unweighted and weighted spatial distribution patterns of the entire stand.The area enclosed by the long dashed line is the conf idence interval;the short dashed line is the image when g(r) is 1;and,the red line is the g(r) image of the test data

Discussion

Weighted and unweighted spatial distribution pattern,and tree size

The aggregation degree of tree weighted spatial distribution patterns was related to the difference of tree size:the more uniform the spatial distribution of the stand,the more the spatial aggregation degree after weighting was related to the difference in forest size,and the more complex the difference in forest size,the more the spatial distribution became more concentrated after weighting.When the stand space was first aggregated,this correlation was weakened,that is,the difference in forest size within the stand cannot be inferred based on the weighted spatial distribution pattern,which was typically related to,for example,intra-or intraforest competition and seed diff usion restrictions.The strong aggregation of seedlings and young trees in the stand was conducive to the survival and development of group effects,and improving the competitiveness of the population and resistance to an adverse external environment.As the individual trees grew,the self-sparing effect and its sparse effect continued to strengthen,which resulted in a large number of forest trees dying,and the aggregation intensity was gradually weakened.In adulthood,a decrease in the intensity of aggregation was beneficial to obtain sufficient environmental resources (Wang et al.2010;Guo et al.2013;Bartels et al.2016;Miao et al.2018).Therefore,young seedlings and saplings in the stands typically aggregated and the larger individuals were randomly distributed (Gu et al.2017;Kris et al.2018).The weighted forest spatial distribution pattern ref elcted the strong interaction between forest trees:the position of the saplings in the physical space was relatively close,but the demand for space resources was still small,and the competing relationship between the trees was weak.After weighting,the spatial distribution pattern aggregation degree was reduced.The spatial distribution pattern of mature forests or tall forests was mostly random,but the demand for space and resources was large,and the interaction force on surrounding forests was strong,so the spatial aggregation degree after weighting was enhanced.Under the condition that the spatial distribution pattern of the stand was uniform or random,the aggregation of the spatial distribution of the forests weighted by the triangulation model could ref lect and compare the complexity of the size diversity of the forest within the stand.The results demonstrated that the change of the un-weighted to weighted spatial distribution pattern was also closely related to the forest ecological succession stage in which the forest is located.

Forest interaction strength and neighborhood topological relationship after weighting

In this study,the approach used to weight the stand space was explored.The purpose of this paper was to fully consider the effects of the DBH,total tree height and crown width of each individual tree in the stand on the growth and development of neighbor trees,which is more realistic.The SSU determined by the triangulation model was based on the fact that each forest tree had a certain“affected area,”and the determination of the central tree neighborhood had certain flexibility (Wang et al.2010;Guo et al.2013).The edge of the triangle in the triangulation model represented the correlation between the two trees.The closer the distance,the more intense the resources and space between the trees.Conversely,the smaller the competition,the more the length of the triangle could be used as a quantitative indicator of the interaction strength between the two trees.The purpose of the weighting in this study was to adjust the distance between the two forests based on the original distance so that the distance between the trees could more accurately characterize the interaction intensity between the two forests,which was essentially the original space of the forest.Based on the distribution pattern,the non-spatial attributes of the forest are transformed into spatial attributes.In addition,the calculation of the attribute weight depends on the weights of the DBH,tree height and crown width.But these weights vary from stand to stand,species to species.Therefore,in this study,the weights of DBH,tree height,and crown width were obtained by Cao’s method,which were 0.36,0.31 and 0.33,respectively,and a weighted triangulation network was constructed.The effects of DBH,tree height and crown width on the attribute weight and the weighted triangulation network need to be further verif ied.

In the triangulation model,the center tree was weighted by its distance from its neighboring trees to form a moving plan equal to the number of neighbor trees.The vector represented the direction and distance of the weighted movement between the center tree and the neighbor tree tree pair.The resulting vector summation determined a logical position relative to the more reasonable center tree of all neighbor trees.The forest weighting changed the position of the forest,which in turn affects each spatial structural unit constituted by trees,that is,the neighbor trees of the target tree were replaced.In this study,the index of the competition index was used to ref lect the interaction between trees.The replacement index of trees neighbor to the target tree was larger than that of the neighbor trees,which indicated the weighted neighboring tree and target tree.The interaction between the two was strong,and the topological relationship between them more reasonably represented the spatial information of the neighborhood of the target tree.The tree topological relationship determined by the weight of the triangulation model was restored to the original forest stand,which provided more scientific theoretical support for the quantitative analysis and optimization of the spatial structure features of the stand (Zeng et al.2007;Li et al.2015;Ghalandarayeshi et al.2017;Riedler and Lang 2017;Zhang et al.2018a,b).

Value of research results

The weighted triangulation network,with its side length,triangle area and Angle between each side,contains the tree’s own attributes,which can better ref elct the actual forest spatial structure.By calculating the area and internal Angle of each SSU,there is sufficient reason to ref lect whether the spatial distribution of trees in the forest is uniform and whether the spatial structure is reasonable,so as to provide help for the sustainable management of the forest.For example,a weighted Delaunay trianglation network is established based on field measurements of a stand to determine whether a single tree is in a reasonable spatial position.When the weighted spatial position of individual trees is obviously unreasonable,for example,small trees occupy too much resource space while large trees have less resource space,then the spatial structure of stand can be improved by means of planting and cutting.In addition,the weighted Delaunay triangulation method adopted in this study can also provide planning guidance for tree planting in order to maximize tree benef tis in limited space.

Conclusion

Based on the SSU determined by the triangulation model,the distance between neighbor trees was weighted.The transformation of the position of the trees expands the analysis method of forest relationship based on distance,makes the expression of the relationship between trees and neighbor trees more indepth and specific,providing an idea for expressing the complex relationship between trees within the forest structure.Correlation analysis between the spatial distribution pattern of weighted trees and tree size standard deviation was performed to further verify the relationship between the weighted forest spatial pattern and the non-spatial attributes of forest stands.The triangulation weighting method indirectly changed the topological relationship between trees in the forest,and the new topological relationship more reasonably resolved the forest neighborhood information.A comprehensive analysis of the differences in unweighted and weighted spatial distribution patterns of forest stands can explain the characteristics of forest ecological succession processes more scientifically.The method based on the triangulation model weighting had greater operability.Compared with the unweighted Voronoi diagram method,it is easy to determine the neighbor tree and calculate the area of the irregular“affected area”.However,because of the differences in the biological characteristics of tree species,the same tree species have similar ecological habits,growth and development processes;hence,the competition for resources and space demand within the species is more intense than competition between species.The shortcoming of this study is that when considering the biological properties of forest trees,only the DBH,tree height and crown width were used as weighting indicators.Future research will consider the tree species and their traits or phylogeny of neighbor trees around the trees,and use them as weights.