Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and Existing Stand Database Information

Kalliovirta, J. & Tokola, T. 2005. Functions for estimating stem diameter and tree age using tree height, crown width and existing stand database information. Silva Fennica 39(2): 227–248. The aim was to investigate the relations between diameter at breast height and maximum crown diameter, tree height and other possible independent variables available in stand databases. Altogether 76 models for estimating stem diameter at breast height and 60 models for tree age were formulated using height and maximum crown diameter as independent variables. These types of models can be utilized in modern remote sensing applications where tree crown dimensions and tree height are measured automatically. Data from Finnish national forest inventory sample plots located throughout the country were used to develop the models, and a separate test site was used to evaluate them. The RMSEs of the diameter models for the entire country varied between 7.3% and 14.9% from the mean diameter depending on the combination of independent variables and species. The RMSEs of the age models for entire country ranged from 9.2% to 12.8% from the mean age. The regional models were formulated from a data set in which the country was divided into four geographical areas. These regional models reduced local error and gave better results than the general models. The standard deviation of the dbh estimate for the separate test site was almost 5 cm when maximum crown width alone was the independent variable. The deviation was smallest for birch. When tree height was the only independent variable, the standard deviation was about 3 cm, and when both height and maximum crown width were included it was under 3 cm. In the latter case, the deviation was equally small (11%) for birch and Norway spruce and greatest (13%) for Scots pine.


Introduction
The development of modern remote sensing sensors has increased the need to create new forest models (Maltamo et al. 2003).One of the most promising methods is to use high resolution digital aerial photographs (Pollock 1996, Gong et al. 2002, Korpela 2004, Wang et al. 2004) or laser scanning (Hyyppä et al. 2001, Holmgren 2003, Naesset 2004) to measure individual trees.As early as the 1970s, Jakobsons (1970) and Talts (1977) described the possibility of measuring the height of a tree, the crown diameter or even the diameter at breast height on aerial photographs by photogrammetry.However, these measurements usually only represent the dimensions of the crown as visible on the aerial photographs, the resolution and visibility of small branches and irregular crown parameters being dependent on the scale of the photograph.In theory, however, a close correlation exists in principle between crown diameter and stem characteristics, such as diameter at breast height, and the latter is also highly correlated with the photogrammetrically measured crown diameter, a relation for which Petlewitz (1976) observed a correlation coefficient of 0.9 in Pinus silvestris and a standard deviation of the regression of 2.5 cm.Klier (1970) emphasized the influence of scale, image quality, species and species mixture, while the close relationship between these variables motivated many researchers (e.g.Sayn-Wittenstein et al. 1967) to construct aerial tree and stand volume tables based on crown diameter.Such tables, based on stand height, crown closure and crown diameter as independent variables or in a modified form (Eid and Naesset 1998, Gingrich et al. 1955, Avery and Meyer 1959), are today in common use in North America and Norway.Krajicek et al. (1961) studied relations of crown and diameter at breast height in open-grown trees not confounded by competition, measuring 340 such trees in eastern Iowa.The crown width of a tree in an open stand is closely related to its diameter at breast height, the correlation coefficient for every species being over 0.98.This relation was found to be independent of age and site quality, but differed slightly between tree species.Open-grown trees were shorter than forest-grown ones of the same diameter on similar soils and under similar conditions.This is attributed to competition between adjacent trees under forest conditions, a factor which also tends to reduce the size of the live crown, and especially the crown width.Ilvessalo (1950) and Jakobsons (1970) studied the correlation between tree crown diameter and diameter at breast height under boreal managed forest conditions.Ilvessalo (1950) found that as branches are cloaked by adjacent trees, measurements of maximum crown diameter on photographs are generally smaller than those made on the ground.Also, crown diameter varies with tree species, tree height, site and stand density.The correlation between crown diameter and diameter at breast height was best for Scots pine and much weaker for Norway spruce.Jakobsons (1970) studied this correlation for pine, spruce and birch separately and reached the following conclusions for trees belonging to the same diameter (at breast height) class.Conifers have smaller crown diameters than deciduous trees, but the location of the tree is also important, such that trees in southern Sweden have greater crown diameters than those in the north.Meanwhile, trees on poor sites or in open stands have greater crown diameters than those on nutrient-rich sites or in denser stands.Jakobsons (1970) also found that an almost linear relation exists between crown diameter and diameter at breast height, although this differed between tree species and between geographically distant trees.The crown diameter of young trees was wider than that of older trees.The relation was also confounded by competition between trees, the availability of light and site factors.Jakobsons (1970) nevertheless maintained that it was possible to estimate diameter at breast height as a function of crown diameter.Talts (1977), by contrast, concluded that also other independent variables in addition to crown diameter were necessary.Nash (1949) and Nyyssönen (1955) found a standard error of 0.6 m in crown diameter estimates on photographs, and Worley et al. (1955) obtained a standard error between 0.9 m and 1.2 m on 1:12 000 photographs.More recently, Hildebrandt (1996) reconstructed the dbh distribution of beech stands from the observed distribution of crown widths.Stand age can also be estimated from a regression equation with photogrammetrically determined stand height and crown size as the predictor variables, although because of the inherent uncertainties, a given stand is usually assigned to one of 20 year classes.Studies in Germany (see Van Laar and Akca 1997) have indicated that the age class of a stand can be estimated from photographic measurements.
New measuring methods, such as laserscanning (Hyyppä et al. 2001, Holmgren 2003, Naesset 2004) or digital photogrammetry (Korpela 2000(Korpela , 2004)); have specific characteristics and measurement techniques.Because imaging condition and applicability of tree measurements differ according to the distance to objects, the relative position of the tree and other similar factors, traditional photography-based crown diameter measurements are not a good basis for modelling.When allometric tree models are created using field measurement, separate calibration models can be used to relate photography-based measurements and ground measurements with improved accuracy.When models are applied directly without calibration using automatic segmentation, small trees are easily overestimated and large trees are underestimated (Ikonen 2004).This type of error can be reduced using calibration techniques which utilize imaging parameters and few field observations (Mäkinen 2004).The models can be directly applied, when laser scanning is used as a remote sensing technique.Tree volume can then be derived from these variables using a chain of models in which diameter at breast height is estimated first.The aim of this study was to investigate the relations between diameter at breast height and maximum crown diameter, tree height and other possible independent variables and to formulate models for estimating the diameter at breast height using different independent variables and chains of models.Models for tree age were also formulated, with height and maximum crown diameter as independent variables.

Material
The main material used in the present work was based on the 1889 permanent sample plots established throughout Finland for the purposes of the Finnish National Forest Inventory (NFI).Plot size varied according to diameter at breast height of a tree.Plot size was 100 m 2 , when diameter was under 10.5 cm and otherwise 300 m 2 .An additional data set (Korpela 2004), comprising 346 Scots pines, 245 Norway spruces and 120 birches on a site near the Hyytiälä Research Station, was used to validate the models.
The NFI sample plot network is based on cluster sampling, where each cluster in southern Finland includes four sample plots and each cluster in northern Finland three.The distance between two clusters is also greater in the north than in the south, as is the sample plot interval.The material contains data from the 1 st and 3 rd rounds of measurements made on the permanent sample plots (in 1985-86 and 1995).
The material includes only trees for which crown diameter measurements are available, and only the data for 1995 were used to formulate the models.The crown diameters in the NFI material were measured according to field instructions, i.e. by taking the widest dimension of the crown.Any obvious mistakes in measuring and recording the data were removed, leaving a total set of 11 246 trees.Trees have been classified according to their position in the stand into the following categories: dominant (63%), intermediate (33%), and suppressed (4%), which refer to determined relative height of tree, over 80 %, 50-80% and less than 50%, respectively.The locations of the clusters are presented in Fig. 1.
The material also includes damaged and diseased trees, which can exhibit a highly abnormal relation between diameter at breast height and either height or crown diameter, causing bias in the models.It is assumed that living trees can be identified by remote sensing material.This may not be the case if the top of the tree is broken or the tree is dying (barely any living canopy left).After removing these abnormal trees, the data used for the diameter at breast height and the age models comprised 5303 Scots pines, 3661 Norway spruces and 2282 birches.The average values for the sample tree and stand variables are presented in Table 1.A caliper was used to measure the diameter at breast height, a Suunto hypsometer to measure tree height, an increment borer to measure tree age and a Kajanus tube to measure crown width.As the relations between tree variables may vary depending on the location (see Jakobsons 1970), the material for the entire country was divided into four geographical areas defined according to the forest flora and climatic conditions (Fig. 1).The resulting distribution is presented in Table 2.

Methods
Due to the hierarchical nature of the data, a mixed effect method with iterative generalized least squares (IGLS) was used for linearized regression.The independent variables were selected according to the requirements defined for the new forest inventory procedure, i.e. that all independent variables should be accessible from high resolution aerial photographs or existing databases.The photogrammetric variables were height, maximum crown diameter, stem number of dominant trees per hectare and relative tree height class.The photogrammetric variables and variables from the stand database were treated as independent variables in the regression model.The intercept was the only fixed effect of the basic model.Clusters and plots were treated as random effects.The form of model is where y is an n × 1 vector of observed values of the dependent variable, b is a p × 1 vector of fixed parameters, X is an n × p matrix of independent variables associated with fixed parameters, c is a q × 1 vector of random parameters with expectation zero, Z is an n × q matrix of explanatory variables associated with random parameters and e is an n × 1 vector of error terms, e ~ N(0,σ 2 ).Furthermore, in this case, k is the cluster to which the tree i in the plot j belongs, c k is the random parameters of cluster k and d kj is the random parameters of plot j.
The variables (α) from existing stand databases that were tested were similar to variables which can be found in the forest planning databases provided by private forest owners in Finland, together with a few generally accepted variables: x co-ordinate, y co-ordinate, height above sea level, temperature sum, mean diameter, mean age, tree class, basal area, land-use class, site class and soil type.Stand variables, which could be derived from an aerial photograph, such as stem number of dominant trees per hectare and relative tree height, were also tested.In general, dominant height is defined as the mean height of the 100 thickest trees at breast height in one hectare.In the context of this study only tree heights can be used to define dominant height because diameters are not known.Dominant tree is defined as a tree which height is more than 80 % from dominant height.Relative tree height could be estimated by comparing the height of the recognized tree to the dominant height of the recognized trees of the remote sensing material on a site.Relative tree height class is used as a dummy variable (D9).It indicates that a tree is suppressed or dominated defined as a tree which height is under 80 percent  from dominant height, therefore differing from a dominant or emergent tree.A model with three variables (h, d crm , α) was chosen for each tree species and area based on a log likelihood ratio test (Goldstein 1995) achieving the best coefficient of determination.
To meet the normality and homoscedaticity assumptions, square root and logarithm transformations were used for the independent and dependent variables.
The models for diameter at breast height were of the forms: (1) and the age models of the forms: where d 1,3 = diameter at breast height (mm), h = height (dm), d crm = crown diameter, maximum (dm) α = stand variable from database or aerial photograph The models were used to estimate the value of the variable in its original unit of measurement.
As non-linear transformations were used for the dependent variables, such an estimate will be biased (Lappi 1993), an effect that can be reduced by bias correction.Taking this into account, the model for diameter at breast height assumes the form and the age model the form R 2 was calculated separately to cluster, plot and tree effects, e.g.R 2 for plot indicates the proportion of variance between plots, that is explained by a model.Proportion of total variance between clusters and between plots are also presented.R 2 was calculated using a method described in Lappi (1997), where relation of estimated full mixed model variance and initial variance of random effect model of clusters and plots (the fixed part includes only a constant) were utilized as follows: The non-linear extra sum of squares method (Bates and Watts 1988) was used to evaluate the differences between the geographical areas.The method requires the fitting of full and reduced models.The full model corresponds to different sets of parameters for each of the geographical areas involved.The reduced model corresponds to the same set of parameters for all regions.
The suitability of the division and the need for any division at all were assessed on the basis of the test results.The appropriate test statistic is described in Bates and Watts (1988).

Data Analysis for Modelling
The normality and homoscedasticity of models were tested.As an example of a model that meets these assumptions well, the diameter of Scots pines at breast height in area 3 is presented in Fig. 2.There were about 2640 pines in the area.Altogether 136 models were constructed.These were numbered using a system in which the first digit for a model defines the geographic area (Fig. 1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last the tree species.For example, model 2.2.3 applies to diameter model for birch in area 2 (tree species = 3), with the maximum diameter of the crown as the only independent variable (form of the model = 2).It should be noted that tree height and maximum crown diameter are expressed in decimetres in all the models, yielding the diameter at breast height in millimetres.

Models for Diameter at Breast Height
The data for all sample plots in the country were used to formulate the first set of models for diameter at breast height.General information on these models is given by tree species in Table 3.As it can be seen, even the best third independent variable, y co-ordinate for Scots pine and temperature sum for Norway spruce and birch, was of minor significance.The models for the diameter at breast height for the entire country are presented in Table 4.
Further models for diameter at breast height were formulated after dividing the data into four geographical areas.General information on these regional models is presented in Table 5.The RMSEs of the models for the ecoregions varied  Table 5. Statistical properties of regional models.R 2 is divided into cluster (Clus), plot (Plot) and tree (Tree) effects.Proportion of total variance (VAR%) is calculated for clusters and plots.The first digit in number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.The third variable for Scots pine in area 1 was the mean age of the growing stock (in years), for Norway spruce the basal area (m 2 /ha) and for birch the mean diameter (cm).In area 2, the third variable for all tree species was relative tree height class (D9).The third variable for Norway spruce in area 3 was the temperature sum (°) and for Scots pine and birch the relative tree height class (D9), while in area 4 it was for Scots pine the relative tree height class (D9), for Norway spruce the basal area (m 2 /ha) and for birch the mean age of the growing stock (in years).The regional models for diameter at breast height are presented in Table 6.

Validation of the Models for Diameter at Breast Height
The functionality of the models was tested with data collected from a site near the Hyytiälä Research Station (in area 2).One aim was to evaluate the convenience of the division into regions, i.e. to determine whether the predicted values differed between the models for the areas and between the models for area 2 and those for the entire country.This implies that the models for area 2 were compared in terms of functionality with those for the other areas, taking into account the differences between tree species.The test results by tree species are presented in Table 7.When evaluating these results, it should be noted that the test data for all models are the same.
The average diameter at breast height for all three tree species is overestimated when the height of the tree is the only independent variable, whereas the models with maximum crown diameter as the independent variable always underestimate the diameter at breast height.When both variables (h, d crm ) are included, the prediction is virtually unbiased.
The average standard deviation when maximum crown width alone was the independent variable was 4.9 cm (about 22% from mean dbh), being smallest for birch.When tree height was the only independent variable, the standard deviation was 3.2 cm, which is about 14% from the mean dbh (smallest for Norway spruce), and when both variables (h, d crm ) were included, it was 2.7 cm (about 12% from mean dbh).The standard deviation for the latter model was equally small for birch and Norway spruce if evaluated in a relative unit of measure, and largest for Scots pine.The third variable models were also tested.In all cases, the effect of the third variable was minor.
The models for the entire country based on the test data predict the diameter at breast height equally well.Only a slight difference existed between the predictions given by the models for the entire country and for area 2, but it is noteworthy that 85% of the trees in the data set for the entire country were located in areas 2 and 3. Had the test data been taken from area 1 or area 4, the differences would undoubtedly have been more marked.
The influence of tree species was studied by comparing models formulated for all tree species with species-specific models.This was done again with the test data from area 2. As might be expected, the latter models predicted the diameter at breast height better than the former, the differences being small for the conifers but considerable for birch (Fig. 3).
The need for ecoregions was tested using the combined model in which the observations from all regions were included.Because the results of F-tests revealed that differences existed among

Models for Tree Age
The models for the age of the tree were formulated with the same procedure as diameter models, using height of the tree or maximum crown width or both as independent variables.General information on the age models for entire country is presented in Table 9, and the models are listed in Table 10.Further age models were formulated for four ecoregions.General information on these regional models are presented in Table 11.The RMSEs of the models for the ecoregions varied between 2.8 and 9.7 years depending on the combination of independent variables and species.Negative R 2 -values in the table indicate that estimated variances may not change logically, e.g. because of correlated regressors.The regional age models are presented in Table 12.
For the all species, the age of the tree was dependent most on its height, and inclusion of the maximum crown diameter increased the coefficient of determination only slightly.For birch, however, the maximum crown diameter was more important independent variable, than for conifers.In some combinations of regions and tree species maximum crown diameter was not statistically significant as independent variable in f(h, d crm ) models.However, the coefficient of determination was quite low in all cases.

Validation of the Models for Tree Age
A validation data set from a site near the Hyytiälä Research Station was also used to evaluate the models and ensure reliability in the prediction for tree age.The growing stock of the site was quite homogenous and only some age measurements were done.So, mean age of the stratifications were used as tree age.This should be noted when evaluating the test results.Table 9. Statistical properties of the age models for the entire country.R 2 is divided into cluster (Clus), plot (Plot) and tree (Tree) effects.Proportion of total variance (VAR%) is calculated for clusters and plots.The first digit in number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.The functionality of the models was different depending on the combination of independent variables and species.For conifers, the prediction of tree age was almost equal when using models, f(h) or f(h, d crm ).Maximum crown diameter as the independent variable seems not to be suitable independent variable of its own.However, maximum crown diameter as the independent variable was the best age model for birch.For Scots pine, it seems that the models for ecoregion 3 were the best although the test site is in area 2. It seems that only height or both height and maximum crown diameter as independent variables for conifers can be used.Maximum crown diameter as the only independent variable worked well for birch.The test results by tree species are presented in Table 13.When evaluating these results, it should be noted that the test data for all models are the same.
The average standard deviation of age when maximum crown width alone was the independent variable was about 30 years (41% from mean age).When tree height was the only independent variable or both variables (h, d crm ) were included, the standard deviation was about 27 years (37% from mean age).For all models, the standard Table 11.Statistical properties of regional age models.R 2 is divided into cluster (Clus), plot (Plot) and tree (Tree) effects.Proportion of total variance (VAR%) is calculated for clusters and plots.The first digit in number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.form and tree species formed exceptions on some pairs of areas.There were only minor differences between the trees species.

Discussion
The primary aim of the modelling was to develop a part of the chain of models required for a new inventory method based on measurements of tree height and maximum crown diameter obtained from high-resolution aerial photographs by digital photogrammetry (Korpela 2000(Korpela , 2004) ) combined with information available from existing stand databases and forest plans.The models could also be utilized when airborne laser scanning data is available.The idea is to predict the diameter at breast height for a single tree by using information derived from aerial photographs and forest plans, which will in turn enable its volume to be calcu-lated.This will mean that the volume of growing stock for a sample plot can be derived from an aerial photograph.Number of independent variables were tested during the study.For example, the number of dominant trees per hectare could be derived from remote sensing data, but it didn't improve the estimation results.According to the tests, the best third variable in the models was basal area.The coefficients of the determination for models with three variables were only slightly better than for those with two variables; thus the benefit achieved with a third variable is negligible.The effect of the third variable was minor also in validation phase of study.Models for predicting the diameter at breast height for a single tree were formulated here based on field data only.Traditionally, aerial photography based volume models are constructed using photogrammetric height and crown width measurements for specific image material.However, the imaging condition and visibility of tree dimensions differ according to the scale of photograph and the relative position of the tree in the aerial photograph.When multiple photographs are utilized, crown dimensions can be measured from several sources, improving the process (Korpela 2004).Laser scanning is one of the most promising technologies in remote sensing-based forest inventories.Stand mean tree height and crown dimensions can be measured relatively accurately from airborne laser scanning data (Hyyppä et al. 2001, Naesset 2004), but further estimation of tree parameters is still required.Mainly these models are planned to be utilised with tree specific procedures, although stand specific procedures could utilise models to estimate mean size of trees.
When allometric tree models are created using field measurements, like in this study, separate calibration models can be used to relate remote sensing-based measurements and ground measurements with improved accuracy.
Because the data set used for modelling contains random measurement errors, the estimated coefficients are biased (Kangas 1998).The statistical tests of the coefficients may also be invalid.However, the coefficients, that are clearly significant remain significant even when measurement errors are taken into account.If the significance is less clear, changes in significance may occur.The effect of random measurement errors on the models can be evaluated by using, for example, the simulation extrapolation method (Carroll et al. 1995).Because no measurement error information is available in the data set, the error effect here is evaluated based on existing studies.The standard error of height using a Suunto hypsometer is, for instance, according to Päivinen (1992) 7.1 dm (3.4%) and Hyppönen and Roiko-Jokela (1978) 8.0 dm (5.7%).No crown diameter measurement error information is available for using the Kajanus tube.If the error of height measurement is assumed to be 5% and the error of crown diameter measurement to 10%, both of which are reasonable, it would be possible to estimate the effect of the maximum error of diameter at breast height.
The models for Norway spruce being the best in terms of RMSE was somewhat unexpected, as according to Ilvessalo (1950), the diameter at breast height can be determined most accurately for Scots pine, the predictions for Norway spruce and birch being much weaker.Scots pines and birches also grow on poor sites, especially on the coast and in northern Finland, where Norway spruce is not found, and seem to produce rather abnormal stem forms there.This could explain the superiority of the Norway spruce models.
The small, young trees (height < 3 m) are a weak point in the models formulated here, and prediction of their diameter at breast height is not necessarily always reliable.On the other hand, these small trees will not be a problem when using the models in an inventory chain if only because they tend to be obscured by the older growing stock in aerial images.An inventory of sapling stand is, of course another matter.The difference in the case of small trees is obviously due to their not having had to compete with adjacent trees for growing space and light, so that the relations between tree variables are slightly different from those for a tree at a later stage of development (Jakobsons 1970).Young trees should therefore have models of their own.Damaged and diseased trees were not included in the modelling.The allometric characteristics do not work well with broken or damaged trees, which mean that these objects should be identified somehow from the remote sensing material.The identification could be based on exceptional allometric features or spectral features in aerial photography.
The applicability and validity of the models was tested with small data set collected from subarea.The conclusion with regard to the modelling of diameter at breast height was the same as that reached by Talts (1977): that crown width is not very reliable as the only independent variable.For example, the heights of the Norway spruces defined the diameter very well, although the crown diameter was not such a particularly good independent variable, at least partly on account of the shaded character of spruces.Tree height was better for this purpose, but it was only when both were used that a reasonable prediction was obtained.This also increased the flexibility of the models, allowing them to take into account the state of competition in the growing stock and its density.Use of models that have at least tree height and maximum crown diameter as independent variables is therefore recommended.To ensure reliability, a division of the country into areas, i.e. regional models, should also be used.
The test results of the models indicated the same.The prediction of tree age proved to be challenging task.For all tree species, the standard deviation of age prediction was large.
The age models were constructed because tree age is an important criteria in defining need for silvicultural treatment.It is important that age estimated are also available in addition to tree size and stand density estimation, when forest information system is used for silvicultural planning.For conifers, the age of the tree was dependent most on its height, and for birch, the maximum crown diameter was the most important independent variable.Relative RMSE of age models for entire country was about 10%.Precision of models was improved significantly when ecoregion specific models were applied.Age prediction for birch was especially difficult.According to the tests, only maximum crown diameter should be used as an independent variable.
Although it is technically possible to measure crown width, crown projection area and crown length on aerial photographs, only the proportion of the crown which is visible can be measured, and the actual maximum crown width can not always be seen because of neighbouring trees.The resolution and visibility of small branches and irregular crown parameters are also dependent on the scale of photograph.One important issue is thus to examine the difficulties encountered in measuring crowns in different stand structures and under varying imaging conditions, involving at least changes in sun-target angle, wind, film and scanning quality.The final estimates can also be affected by local topographical variation.Thus, numerous factors can potentially cause error in photogrammetric forest inventories.The models might behave wrongly when those are applied with unexpected combination of independent variables.Still, the modelling data set is covering entire area of Finland and measurement of permanent sample plots of NFI are carefully collected, which should ensure that most of existing variation of target area is modelled properly.However, the models constructed here serve the need to estimate tree characteristics from crown dimensions from different remote sensing materials and will reduce the need for fieldwork in single tree-based forest inventory procedures.

Fig. 1 .
Fig. 1.Models were constructed for all of Finland (right side) and for four separate regions (left side).Geographical regions are defined by the forest flora and climatic conditions (1 = Hemiboreal, 2 = South boreal, 3 = Middle boreal, 4 = North boreal).The entire area is covered by clusters.The locations of the clusters are shown on the right side of the figure.

Fig. 2 .
Fig. 2. Diagnostic testing of the model d 1,3 = f(h, d crm ) for Scots pine in area 3. Residual plot in the left side and normality plot of residuals in the right side.

Fig. 3 .
Fig. 3. Averages and standard deviations for predicted values of d 1,3 = f(h, d crm ) in models for area 2 with and without information on tree species.

Fig. 4 .
Fig. 4. Interpolated residual surfaces obtained from the dbh models for Scots pine, Norway spruce and birch formulated over the entire country (left side) and for the four geographical areas (right side).

Table 1 .
Mean statistics of field material (NFI) by species.
* D crm refers to maximum crown diameter

Table 2 .
Number of trees of NFI field plots in different geographical areas.

Table 3 .
Statistical properties of the models for the entire country.R 2 is divided into cluster (Clus), plot (Plot) and tree (Tree) effects.Proportion of total variance (VAR%) is calculated for clusters and plots.The first digit in number of model refers to the geographic area (Fig.1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.

Table 4 .
Parameter estimates and t-test statistics (t) of models for diameter at breast height for the entire country.The first digit in number of model refers to the geographic area (Fig.1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.

Table 6 .
Parameter estimates and t-test statistics (t) of regional models for diameter at breast height.The first digit in number of model refers to the geographic area (Fig.1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.

Table 8 .
F-tests of the regional differences of diameter models: d=f(h, d crm ) by tree species.

Table 10 .
Parameter estimates and t-test statistics (t) of the age models for the entire country.The first digit in number of model refers to the geographic area (Fig.1) in question (number of the area or 9 as an indication of the entire country), the second digit the form of the model and the last digit the tree species.

Table 14 .
F-tests of the regional differences of age models: Age=f(h) by tree species.