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The spatial distribution of trees in forests can be described and modelled by point processes where the points are given by the locations (coordinates) of the trees. Further properties of a tree like height or mean crown radius can be interpreted as so called marks of the considered point process characterising the points or trees in some way. The so called mark correlation function describes the spatial correlation of these marks in the observed point pattern. In this paper we introduce a special mark, the overlapping or crown index. We show that mark correlation functions for the considered marks help to understand interaction effects of forest trees.

Generally, forest scientists as well as wood processing industry are interested in spatial dimensions and the distribution of trees in stands. The spatial distribution of trees as well as the dimension of a tree like height and mean crown radius depends on interaction and competition effects in forests. Obviously, interaction among trees results in certain tree heights and crown radii in specific neighbourhood situations with trees of one or more species. Think about the following situations: An old tree suppresses young and small trees of other species under his crown, because for these young trees necessary resources like water and light are hardly available. In some cases contrary interaction effects are conceivable. For example, surrounding trees of a certain species support each other with respect to growth and height even in order to prevent trees of other species in their region.

Mixtures of tree species are in focus of forest ecosystem related research for some years now (see

From a statistical point of view we interpret a forest or stand as a point process. Obviously, the points are given by the positions,

In ecological statistics especially with respect to forestry point process models are often used to analyse forest dynamics and even regeneration in forests, see (

Sometimes additional information should be taken into account. For the stand discussed in this paper for every point or tree so called marks are available characterising the tree in some way. Of course, the height and the mean crown radius of a tree are such marks. Additionally, we introduce a further mark: the so called crown or overlapping index. This mark measures to what extent the crown of the corresponding tree lies under crowns of surrounding trees. Analysing this mark or index can help to describe and understand specific neighbourhood situations or tree-tree mixtures with respect to several species. In general, indices quantifying spatial forest structure are frequently used to monitor spatial aspects of tree characteristics including biodiversity in research plots of limited size.

Besides the spatial distribution of marks or indices the spatial correlation of marks is of great interest even with respect to interaction effects among trees. This spatial correlation of marks can be analysed using the well-known mark correlation function, see (

In this paper we discuss the spatial correlation of the given and determined marks for a mixed stand of European beech and Norway spruce. To distinguish between interaction effects among trees of the same species (intra-specific interaction) and interaction among trees of different species (inter-specific interaction) the mark correlations functions are calculated for beech-beech and spruce-spruce mixtures as well as for mixed spruce-beech (or beech-spruce) situations.

Our study site is located in the Solling area. The Solling is an intermediate mountain range in the south of the German Bundesland Lower Saxony. The studied forest is located at 400 m a.s.l. at 51°43´N and 9°38´E. In this region and elevation annual average precipitation of 900 L/m² and annual mean temperature of 7.5°C are reached. During the vegetation period (May to September) the averages are 420 L/m² and 13.4°C mean temperature. The soil is a dystric cambisol (FAO classification) from loess solifluction overlying sandstone. Within the Solling the experimental plot is located in a 130 to 140 years old European beech

For every tree the crown radius was determined for 8 directions anticlockwise from 0° to 315° (45° steps). These 8 values lead to the mean crown radius which is calculated as arithmetic mean of these 8 values. See

Additionally, the height of each tree is available and the overlapping index will be calculated considering spruce-spruce-, spruce-beech- and beech-beech-compositions.

Let _{1} _{n} . At every point _{i} a so called mark _{i} , i=1, …, n, that means a value characterising the point in some way, is measured or determined. In our case the points _{i} represent the positions of the trees in the experimental plot,

The spatial correlation of marks in this marked point process is described using a so called test function _{i,} _{j} ) for two marks m_{i} and m_{j}. This test function characterises the correlation between the marks _{i} and _{j} at different points, conditional on the distance h between the points. Here, we use (

In general, the mark of a tree is a random number. This means that the test function is random, too. Obviously the test function given above leads to large values, respectively small values, if large, respectively small, non-negative marks, are given. Therefore, this test function indicates correlation between marks in a natural way. From point process theory further test functions are known, which may all result in similar ecological interpretations (_{i} _{j} _{i} _{j})^{2} leads to the so called mark variogram familiar with the well-known variogram from geostatistics which is also often applied in forestry statistics, see (

Now, let _{i} _{i} _{j} _{i} _{j} is given by (

Once again, the test function is a random number and therefore its mean is well defined. The numerator in

Obviously, if the marks are uncorrelated we obtain _{f}

Now, from an ecological point of view it is of great importance to connect values of the mark correlation function with interaction effects like attraction, avoidance behaviour or inhibition of trees.

Negative correlation standing for inhibition and suppression with respect to the classical correlation coefficient is indicated by _{f} _{f} _{f}

Frequently, the marks are positive numbers (tree height, dbh, etc.).

In this case the test function _{i} _{j} _{i} _{j} always leads to positive values of the mark correlation function. Therefore, it is not possible to interpret values of the mark correlation function directly as correlation coefficients. But the criteria for positive and negative correlation are given above. Therefore, choosing the test function _{i} _{j} _{i} _{j} leads to an easy identification of the type of correlation.

The mark correlation function from _{i} _{j} _{i} _{j}

where _{i} _{j} with distance

Usually, estimated mark correlation functions from

Therefore, it is useful to use kernel estimators to obtain smoother functions. Let us mention the so called Epanecnikov kernel (

where

Using this kernel we obtain a smooth estimate of the mark correlation function by (

Note that the points _{i} _{n} , _{i} _{i} _{i} _{i} _{i} _{i} _{j} respectively, leads to the window _{i} _{j} respectively, where

For numerical calculations we used

The mark correlation function is an important tool to monitor spatial variability of marks of trees as well as interaction effects of the corresponding trees. “Classical” marks of trees such as tree height and dbh are not sufficient to describe the spatial structure of a stand with respect to forest dynamics and regeneration processes because the availability of resources like water and radiation strongly depends on the canopy structure even characterized by overlapping crowns of trees. Therefore, to analyse neighbourhood situations of forest trees with respect to the canopy structure of the stand we introduce the so called overlapping or crown index. For every tree crown radii are given for 8 directions. Let us denote a basal point at the end of a crown radius as

Now, we can prove if a crown point of a fixed tree lies inside the crown surface (octagon) of another tree. This check is done in the following way: at the first step we calculate the crown area of a tree as area of the corresponding crown surface (octagon). At the second step we calculate for every crown point of the remaining trees the area of the polygon generated by the 8 original crown points of the fixed tree and this additional crown point. If this crown point lies within the crown area of the fixed tree, then the area of the 9-point-polygon is smaller than the original crown area. In the opposite case,

Let _{1} _{n}

Now, the crown index for the tree at

Obviously, if a solitary tree without crown points under crowns of other trees is given at s we obtain _{CI} _{CI}

The crown indices for different mixtures of trees are given in

Now, we want to have a look on the mark correlation functions. Firstly, let us discuss the mark correlation function for the mark tree height, see

Different situations are given if mixtures are analysed considering surrounding trees from one species. At small distances we obtain positive correlation for the beeches and negative correlation for the spruces.

In comparison with

Therefore, in the case of spruce-spruce mixtures we have attraction at small distances for the mark mean crown radius but negative correlation or suppression for the mark tree height. Spruce-spruce mixtures are especially given in the north-western part of the stand. The results for the two mark correlation functions describe the fact that spruces with different heights are relatively often given at small distances. Now, in the case of beech-beech mixtures we have attraction at small distances for the mark tree height but negative correlation for the mark mean crown radius. This corresponds to the observation that there are some situations in the stand with beeches of similar height but overlapping or interacting crowns, see

Finally, let us discuss the mark correlation function for the mark crown index, see

At distances smaller than 45-50 metres negative correlation is given for the beeches and spruces. Within these distances the mark correlation function for the beeches leads to smaller values than the mark correlation function for the spruces. This confirms our results for the other mark correlation functions from

Obviously, there is interaction among trees in forests. From an ecological point of view it is important to distinguish between intra-specific interaction corresponding to trees from one species and inter-specific interaction concerning two or more species. Of course it is possible to model the spatial distribution of trees directly by using a point process model with an intensity function taking into account interactivity, see for example (

In this paper we present an alternative approach. Sometimes characteristic features of trees like tree height and mean crown radius are available. The spatial distribution of such features or marks of the corresponding spatial point process of tree positions also provides information about interaction in the stand. This information can be measured by the corresponding mark correlation function. The additional mark crown index introduced in this paper helps to understand specific types of inter- and intra-specific interaction. Especially, the spatial structure of the stand can be analysed in a reasonable way. In the study site the spatial structure of the mixed stand is characterized by groups of beeches and spruces in the following way: Many overlapping situations of the crowns of the beeches occur. Similar heights of neighbouring beeches require this behaviour of the beeches. Neighbouring spruces tend to have different heights. Nevertheless, overlapping situations of crowns happen more rarely. It is remarkable for a stand without silvicultural interferences in the last years that interactions between crowns of different species occur very rarely.

Our study has shown that the mark correlation function seems to be a simple tool for indicating the spatial structure in a stand.

With this paper we want to contribute to the broader dissemination of modelling interaction using the mark correlation function in forestry and ecological research. We are convinced of the applicability of this approach in many ecological studies. The simplicity of such approaches connected with the applicability guarantees the chance to use the mark correlation function in various ecological situations and studies.

This research was supported by grants from the Deutsche Forschungsgemeinschaft (DFG). The authors are grateful to W. Näther for useful discussions and to N. Frischbier, who measured the crown radii for the 254 trees.

^{th}edn). Academic Press Inc., London.

European beech (o) and Norway spruce (+) with contour lines of the crowns (crown areas).

Mark correlation function indicating positive correlation (dotted line), negative correlation (dashed line) and independence (solid line).

Mark correlation for tree height: Solid line: beech-spruce situations; dotted line: spruce; dashed line: beech.

Mark correlation for crown radius: Solid line: beech-spruce situations; dotted line: spruce; dashed line: beech.

Mark correlation for crown index: Solid line: beech-spruce situations; dotted line: spruce; dashed line: beech.

Characterisation of the beech-spruce mixed stand. Data for the study site. N: sample, Mean: mean, STD: standard deviation. Crown area: basal area of the crown generated by the crown contour lines. DBH: diameter at breast height.

tree | N | Height [m] | DBH [cm] | Crown area [m²] | |||
---|---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | ||

beech | 154 | 25.70 | 4.93 | 39.82 | 11.83 | 66.67 | 4.1 |

spruce | 84 | 35.01 | 3.77 | 63.51 | 11.96 | 41.95 | 2.54 |

total | 238 | - | - | - | - | - | - |

Crown index for inter-specific (spruce-beech) and intra-specific (spruce-spruce and beech-beech) situations. Mean: mean, STD: standard deviation.

Crown index | intra-specific | inter-specific | ||
---|---|---|---|---|

Mean | STD | Mean | STD | |

beech | 0.32 | 0.30 | 0.85 | 0.24 |

spruce | 0.68 | 0.31 | 0.85 | 0.24 |