Volume 7
Issue 2
Forestry
JOURNAL OF
POLISH
AGRICULTURAL
UNIVERSITIES
Available Online: http://www.ejpau.media.pl/volume7/issue2/forestry/art04.html
EMPIRICAL FORMULAE TO ASSESS THE BIOMASS OF THE ABOVEGROUND PART OF PINE TREES
Jarosław Socha, Piotr Wężyk
The material for the studies were the results of the measurements carried out on 113 experimental trees coming from the experimental plots located in the monitoring network established in the Niepołomice Forest (Puszcza Niepołomicka) within the project FOREMMS 5PR UE [20]. After cutting down the trees wet biomass of twigs with needles was determined. Stem discs were also taken to make stem analysis. In laboratory conditions dry mass of twigs with and without needles was determined. The dry mass of stem and twigs was determined based on wood density in pine. Using the correlation analysis the features explaining most variability of respective components of the aboveground biomass of trees were selected. These features were used to assess empirical formulae to measure biomass. The formulae to assess biomass of stem, twigs, needles, cones were made as well as the formula to measure the whole biomass of the aboveground part. At the present stage the proposed empirical formulae can be applied
Key words: needles biomass, twigs biomass, stem biomass, allometric formulae, Scotch pine..
Biomass is becoming more and more often seen as future renewable energy source [19] and applied, among others in the biofuel production [8]. The studies on biomass make a valuable source of information on the complete content of carbon and nutrients in a forest ecosystem and are used in the modeling of the carbon circulation in nature [6]. To define the amount of the biomass production usually empirical formulae are applied. These formulae were applied for selected forest tree species [2, 5]. Due to the way of the tree market the majority of used nowadays empirical formulae were made regarding the volume rather than the weight of wood [4]. In the case of wood, stem or twigs the knowledge of woods volume and density allowed to define its biomass. It is slightly more difficult to define the biomass of assimilation apparatus. Apart from numerous formulae to define the volume of respective components of the aboveground part of the trees, formulae to assess the biomass in weight units are creat ed. In Czechoslovakia empirical formulae to define the biomass of assimilation apparatus and selected parts of the aboveground biomass of young pine tree complexes were elaborated by Chroust [7]. Other studies on the size of aboveground biomass of trees were done by Mäkelä and Vanninen [16] as well as Vanninen et al. [19]. The size of the biomass of assimilation apparatus of pine was studied by (among others) Albrektson [1]. In Poland detail studies on the formation of the biomass of assimilation apparatus of Scotch pine (Pinus sylvestris L.) was carried out by Lemke [10, 12, 13] as well as Lemke and WoĽniak [15]. Their result was making the empirical formulae to assess the biomass of needles and needled twigs. Lemke also carried out the studies on the volume of pine twigs [11]. Formulae to asses the biomass of the respective components of aboveground parts of trees for 12years pines of provenience experiments were made by Oleksyn and his coworkers [17]. The size of the biomass of assimilation apparatus of naturally renewed pines in younger age classes was investigated by Barcikowski and Loro [3].
The results of the studies by Koeper and Richardson [9] indicate that empirical formulae elaborated for a concrete area when applied in another area can sometimes generate considerable errors. Thus the definition of biomass production for a concrete area demands work consuming and costly measurements of production in local conditions.
The goal of the presented studies was to elaborate empirical formulae to assess the biomass of respective components of aboveground parts for pine trees from the Niepołomice Forest.
The material for the studies are the results of measurements carried out in July 2001 on 116 experimental trees. They came from randomly situated 185 experimental plots localized in a regular monitoring network  750x750m (fig. 1), established in the Niepołomice Forest within the project FOREMMS 5PR UE [21].
On each experimental plots one sample tree was cut down. Its height and breast height was selected in such away to be the closest to average dimension in each tree group. The number of trees in respective classes of thickness was presented in table 1.
Fig. 1. Regular grid of the monitored surfaces FOREMMS in the Niepołomice Forest 
Table 1. The distribution of trees in diameter and height classes 
Height (m) 
Diameter at breast height (cm) 

Total 

6 
8 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
48 
56 

4 
1 




















1 
6 
2 
1 



















3 
10 


1 


















1 
12 



2 

















2 
14 


2 
3 


1 














6 
16 




1 
















1 
18 




1 
1 
1 

1 












4 
20 





1 
1 
1 
1 

1 










5 
22 






4 
3 
3 
2 
3 
1 


2 






18 
24 







2 
1 
3 
5 
5 
4 
3 
1 
2 





26 
26 









1 
5 
4 
6 
3 
3 
1 
2 
1 



26 
28 












2 
4 
7 


3 

1 
1 
18 
30 












1 
1 
1 
1 





4 
32 


















1 


1 
Total 
3 
1 
3 
5 
2 
2 
7 
6 
6 
6 
14 
10 
13 
11 
14 
4 
2 
4 
1 
1 
1 
116 
The age of analyzed trees ranged from 7 to 148 years, breast height diameter ranged from 5.4cm to 55.3cm and their height was from 4.95m to 32.36m. After cutting down respective trees their length and the length and wideness of the crown were measured. The crown of each tree was divided into twigs with and without needles (with cones). The mass of branches was determined in the field with the scales of the accuracy up to 0.5 kg. In the case of twigs with needles the accuracy of the measurement was 0.001 kg. From the collected biomass of twigs with needles for further analyzes a several kilogram sample was taken for each tree. Then in laboratory conditions, after drying in a hypopressure dryer (60°C) and the division into fractions, dry biomass of twigs with needles, needles and cones was measured. The establishment of weight proportions of respective fractions allowed the calculation of the total biomass of twigs with needles, needles and cones for the trees. On each logged tree full s tem analysis was carried out. To do this on the heights of 0.0m, 0.5m, 1.3m, 2m, 4m and then every two meters up to the summit of the stem discs for the analysis were taken. Based on stem analysis features as (among others stem volume (V) (with and without bark), volume increment (I_{V}) and height increment (I_{H}) of trees were determined. On the disc from the height 1.3 m the value of the breast height increment (I_{D}) was determined as well as the the basal area increment (I_{BA}) in subsequent years of the life of trees. The biomass of the wood from the stem in the dry condition (B_{s}) was defined by the conversion of volume calculated based on stem analysis into weight units. To do this the data concerning the density of wood and bark in pine were used. Based on the studies by Wagenführ and Scheiber [20] it was assumed that the, mean density of pine wood in dry condition is 490 kg/m^{3}, while the density just after logging 820 kg/m^{3}. The dens ity for bark for pine is 300 kg/m^{3}.
The biomass of branches (B_{b}) was defined by the multiplication of wet biomass and the ratio between dry wood density and wood density after logging.
(1) 
where:
B_{d} – dry biomass of branches (kg)
B_{w} – wet biomass of branches (kg)
g_{d} – density of pine wood in dry condition (490 kg/m^{3})
g_{w} – density of pine wood just after logging (820 kg/m^{3})
The biomass of bark was determined based on the volume of bark, calculated for the stem and its density in dry condition.
For the description of the relationships between the biomass of subsequent components of aboveground parts of trees and selected explanatory variables an allometric equation [18] of general form was applied:
(2) 
where:
Y – dependent variable
X_{1}, X_{2}, ... X_{n} – independent variables
b_{0}, b_{1}, b_{2}, ... b_{n} – parameters of the equations
To obtain homoscedasticity of the residuals (constant of residual variance for the whole range of the biomass value), parameters of equations for the definition of the biomass of stem and branches were estimated after the transformation of allometric function by finding a logarithm (formula 3).
(3) 
Final selection of explanatory variables in the equations to define biomass resulted from the participation explained variance, defined based on the adjusted coefficient of determination. For subsequent equations the analysis of residuals was carried out where the occurrence of autocorrelation was examined and the distribution of residual values in the relation to the values predicted according to the equations was analyzed.
The assessment of the accuracy of empirical formulae was carried out on the collected study material. This was based on the calculation with the created formulae the size of subsequent components of aboveground biomass of trees and comparing them with the actual value, assumed to be a measured value. The accuracy of respective formulae was characterized with absolute errors and percentage errors the values of which were described with the values of mean, standard deviation and extreme values (negative and positive).
Based on the carried out analyses empirical formulae to define the biomass of respective components of aboveground parts of the tree were created. Taking into account the possibility of practical application of the created formulae, apart from the solutions of the greatest accuracy (the greatest participation of explained variance) slightly less accurate solutions were also proposed, because they demand a smaller number of explanatory variables.
1. Formulae to define stem biomass
Dendrometric features that to highest degree explain the variability of stem biomass are breast height diameter and height of the tree. The application of extra explanatory variables such as: relative crown length or the breast height basal area increment did not cause the increase of the participation of explained variance. Finally the empirical formula to define stem biomass (B_{s}) contains two explanatory variables: height (H) and breast height diameter (D).
(4) 
where:
B_{s} – means stem biomass
H – tree height
D – breast height diameter
b_{0, }b_{1}, b_{2} – parameters of the formula
ε – random error
The parameters of formula (4) to define dry biomass of stem with and without bark are put in table 2. Based on the corrected coefficients of multiple correlation it can be stated that the elaborated formulae explain above 99 % of the variability the biomass of stem with and without bark (tab.2). residual analysis carried out with DurbinWatson test did not show the occurrence of the autocorrelation of residuals. Both in the case of the formula on the biomass of stem with bark and the formula on the biomass of stem without bark a symmetric distribution of residual values in the relation to predicted values was found (fig.2).
Table 2. Parameters and basic statistical characteristics of empirical formulae for the definition of stem biomass 
B_{s} 
Parameters of equation 
R 
R^{2} (adjusted) 
pvalue 
Durbin Watson Statistic 
Serial correlation 

b0 
b1 
b2 

Stem with bark 
0.015402 
1.14754 
0.94829 
0.9961 
0.9921 
0.0000 
1.923427 
0.038031 
Stem without bark 
0.012016 
1.242979 
0.931534 
0.9957 
0.9913 
0.0000 
2.001216 
0.01035 
Fig. 2. The distribution of the residual values of the stem biomass with bark (a) and stem biomass without bark (b) (for the data converted into logarithm) compared with the values predicted according to equation 1 
2. Formulae to define the biomass of branches
In the case of the fraction of branches (without needles) it was found that a relatively high accuracy in the biomass determination can be obtained with the use of two explaining variables (like in stem), e.g. breast height diameter and tree height (formula 5, tab. 3).
(5) 
where: B_{B} – means the biomass of branches
The accuracy of the empirical formula to define branch biomass is greater, because to define it additionally the growth of the surface of breast height section of the tree is included (formula 6, tab. 3).
In the case of the application of two basic variables, e.g. breast height diameter and the height of the tree, the formula explains about 81% of the variability in the biomass of branches (tab.3). Including the breast height basal area increment of the tree causes the increase of the participation of explained variance up to about 86%. Based on the analysis of residuals in formulae 5 and 6 no statistically significant autocorrelation of residuals was found. Distribution of residual values compared to the predicted values based on the equations is symmetric towards X axis. The range of residuals is much smaller in the case of equation 6 (fig. 3 b), where the biomass of branches is defined based on: height, breast height diameter and 5 years’ basal area increment. Extreme value of errors get much smaller in this case (compare fig. 3a and 3b).
(6) 
where: I_{BA5}  means 5years’ breast height basal area increment
Table 3. Parameters and basic statistical characteristics of empirical formulae for the definition of dry and wet biomass of branches 
No 
B_{x} 
Parameters of Equation 
R 
R^{2} (adjusted) 
pvalue 
Durbin Watson Statistic 
Serial correlation 

b_{0} 
b_{1} 
b_{2} 
b_{3} 

2 
Branches 
0.127454 
0.859646 
1.276049 
0.9041 
0.8141 
0.0000 
2.074397 
0.045494 

3 
2.199512 
0.759162 
1.122350 
0.417785 
0.9272 
0.8560 
0.0000 
2.041517 
0.024747 
Fig. 3. The distribution of the residual values of branches biomass (for the data converted into logarithm) compared with the values predicted according to equation 5 (a) and equation 6 (b) 
3. Formulae to define the biomass of assimilation apparatus
The biomass of assimilation apparatus in Scotch pine can be defined with certain accuracy using empirical formulae based only on two basic dendrometric features of the tree, e.g.: breast height diameter and height (formula 7).
(7) 
where: B_{dfb} – dry biomass of needles
The coefficient of correlation for the given relation (7) is 0.83. Defined based on a corrected coefficient of the determination of the participation of explained variance is 73.0%. Large increase of the participation of explained variance (the increase of accuracy of the prediction of needles biomass) can be obtained with the use of the formula, where apart from breast height diameter and height of the tree an explanatory variable is also the growth of the surface of breast height section (formula 8, fig. 4).
(8) 
In the case of the application of the breast height basal area increment the correlation coefficient reaches the value of 0.90 and a adjusted determination coefficient reaching 0.81 shows that the created formula explains as much as 81% of the variability of needle biomass.
Fig. 4. The distribution of the residual values of the needles biomass (for the data converted into logarithm) compared with the values predicted according to formula 8 
4. Biomass of cones
The biomass of pine cones from the area of Niepołomice Forest is a feature that was very variable. I the case of many younger trees cones were totally absent. For the whole sample the biomass of cones ranged from 0.11kg to 9.50 kg. It was stated that only breast height diameter of the tree was significantly correlated with the value of this feature (r=0.63). Thus for the biomass of cones (B_{c}) it was only possible to give an approximate relationship (formula 9), for which the participation of explained variance was about 38.9%.
(9) 
Regarding very high fluctuations in the number of cones, connected with seed years and other factors influencing their size, one can hardly assume that it would be possible to create an empirical formula allowing exact determination of cone biomass. The relation given above can only be used to estimate biomass, allowing very big errors.
5. The formula to define total aboveground biomass of the tree
To define total biomass of all the aboveground parts of the tree a formula where explanatory variables are breast height diameter and tree height (formula 10).
(10) 
The coefficient of multiple correlation for the above formula is 0.9918, and adjusted coefficient of determination is 0.9836. Thus the created formula explains about 98.4% variability of aboveground tree biomass. DurbinaWatson’s test showed that the application of the presented formula construction does not cause the autocorrelation of residuals. Based on the distribution of residual values in the relation to the values predicted by equations, one can state that the application of the formula to define a total tree biomass does not impose systematic errors in the definition of the volume, related to the value of tree biomass (fig. 5).
Fig. 5. The distribution of the residual values of the tree biomass (for the data converted into logarithm) compared with the values determined by the equations 
6. The assessment of the accuracy of empirical formulae
Mean absolute errors in determining the biomass are usually in the case of respective fractions small and range from – 5.54 kg for total tree biomass to – 0.40 kg for needles biomass defined from formula 8 (tab. 4). Mean values of percentage errors are slightly above zero. The highest values of mean percentage errors were obtained in case of the formulae to determine the biomass of cones, branches and needles. Mean errors in the determination of stem biomass and tree biomass are close to zero. The range of percentage secondary errors in defining the biomass of branches was in case of formula based on breast height diameter and height was from – 53.3% to 302.7%, after including five years’ breast height basal area increment diminished and was from – 54.5% to 120.8%. Also in case of the biomass of needles the application of the basal area increment of the tree caused considerable increase of the range of percentage errors and standard deviation of percentage errors dimin ished from 37.12% do 26.44%. Cones biomass is very little correlated with biometric elements of the tree, thus the formula to calculate it, where the only explanatory variable is breast height diameter of the tree can generate big errors ranging from – 89% to 1060%.
Table 4. Characteristic of the accuracy of the created empirical formulae to define the biomass of respective components of aboveground parts of pine 
Fraction of Biomass (formula) 
Absolute Error 
Secondary Percentage Error 

Mean 
Standard Deviation [kg] 
Extreme Error 
Mean 
Standard Deviation [%] 
Extreme Error 

negative 
Positive 
Negative 
Positive 

Stem Biomass with Bark (4) 
2.62 
49.32 
150.7 
201.4 
0.18 
9.53 
21.3 
24.3 
Stem Biomass without Bark (4) 
2.54 
46.71 
141.9 
171.7 
0.24 
10.22 
24.2 
29.5 
Branches (5) 
2.89 
18.93 
97.4 
43.3 
7.33 
46.92 
53.3 
302.7 
Branches (6) 
2.01 
19.66 
90.6 
84.1 
5.16 
34.50 
54.5 
120.8 
Needles (7) 
0.41 
4.26 
13.0 
10.8 
6.75 
37.12 
55.1 
173.6 
Needles (8) 
0.40 
3.94 
12.6 
9.7 
3.40 
26.44 
53.4 
84.6 
Cones (9) 
0.42 
1.72 
8.1 
3.2 
52.23 
172.08 
89.9 
1060.9 
AboveGround Tree Biomass (10) 
5.54 
63.46 
184.7 
288.8 
0.45 
12.69 
27.1 
37.9 
Elaborated empirical formulae are designed to estimate the value of the biomass of the aboveground part of the tree for Scotch pine. Due to the origin of the material for the studies, the equations presented at this stage can only be used locally for the tree stands of the Niepołomice Forest. In the case of their application outside of the study area the obtained results can have serious systematic errors. Practical application of the created formulae on a wider scale demands the assessment of their accuracy on an independent empirical material coming from different habitats and regions of the pine’s range. To determine the biomass of the wood of the stem the wood densities found in the literature [20] were applied. The determination of wood density in local conditions can allow the correction of the proposed formulae. The correction should mean the multiplication of the result obtained from the given formulae by the ratio of actual density to the density applied in the presented research. The analysis of the size of errors in respective formulae allows to state that the possibility of the exact definition of the biomass of respective components for single trees is limited. Already in the case of the trees used to make the above formulae, the errors sometimes reached big values. Large percentage errors in the measurement of the biomass can be expected particularly in the case of the following fractions: branches, needles and cones e.g. features characterized by a large variability. The biomass of stem and all the aboveground part of the tree can be defined with much smaller percentage error. Smaller errors in the definition of the size of biomass can also be expected in the case of the application of the elaborated formulae in measuring the biomass of the larger sample of trees.
SUMMARY OF RESULTS AND CONCLUSIONS
In the framework of the studies carried out in the Niepołomice Forest empirical formulae to define the biomass of stem, branches, needles, cones and total aboveground biomass of Scotch pine were created.
The biomass of pine stem and biomass of aboveground parts of the tree can be defined with a relatively great accuracy based on the equations where explanatory variables are breast height diameter and tree height. The accuracy of prediction determined based on the participation of the explained variance was above 99%. the application of additional biometric features of the tree (including the length and width of the crown) did not increase the accuracy of the prediction of stem biomass.
The biomass of pine branches can be determined based on allometric formulae where explanatory variables are breast height diameter and tree height. The participation of explained variance is about 90%. The increase of the accuracy in measuring the biomass of branches can be obtained by the application of an additional explanatory variable in the form of the breast height basal area increment of the tree. This causes the increase of explained variance by about 2%.
The estimation of the biomass of needles using empirical formula, where explanatory variables are breast height diameter and height in the case of single trees can cause considerable errors. Much higher accuracy in the determination of the biomass of needles can be obtained after the application of an additional explanatory variable in the form of the breast height basal area increment. The participation of explained variance grows from 73% to 81% in this case.
Because of a large variability and seasonal fluctuations, exact estimation of the biomass of cones is impossible. Approximate information on cone biomass can be obtained by the application of an equation where the only explanatory variable is breast height diameter. The participation of explained variance is only 38.9%, though.
To define the total biomass of aboveground part of the tree an empirical formula explaining above 98% of its variability was created. Application of the formula requires the knowledge of two basic features of the tree: breast height diameter and height.
At the present stage of the studies the created empirical formulae are recommended only locally for the pine tree stands of the Niepołomice Forest. Their practical application of larger scale should follow checking the accuracy on an independent empirical material coming from different places and different types of habitat.
Albrektson A., 1984. Sapwood Basal Area and Nedle Mass of Scots Pine (Pinus sylvestris L.) Trees in Central Sweden. Forestry, Vol. 57, No.1. 3543.
Baker T.G., Attiwill P.M., Stewart H.T.L. 1984. Biomass equations for Pinus radiata in Gippsland, Victoria. New Zeland Journal of Forestry Science. 14 (1), 8996.
Barcikowski A., Loro P.M. 1995. Needle biomass and dendrometrical features of Scots pine (Pinus sylvestris L.) natural regeneration seedlings of younger age classes, growing on a fresh coniferous forest site. Sylwan 139(2): 5362. [in Polish]
Bruchwald A., RymerDudzińska T., Dudek A., Michalak K., Wróblewski L., Zasada M. 2000. Wzory empiryczne do okre¶lania wysoko¶ci i pier¶nicowej liczby kształtu grubizny drzewa. [Empirical formulae for defining height and dbh shape figure of thick wood]. Sylwan, 10: 513. [in Polish]
Eliott K.J., Clinton B.D. 1993. Equations for Estimating Biomass of Herbaceous and Woody Vegetation EarlySuccessional Southern Appalachian PineHardwood Forests. USDA Forest Service. Research Note SE365.
Grote R. , 2002. Foliage Branch Biomass Estimation of Coniferous and Deciduous Tree Species. Silvia Fennica 36(4): 779788.
Chroust L. 1985. Aboveground biomass of young pine forests (Pinus sylvestris) and its determination. Communicationes Instituti Forestalis Cechosloveniae 14: 127145.
Johansson T., 1999. Biomass Production of Norway Spruce (Picea abies (L.) Karst.) Growing on Abandoned Farmland. Silva Fennica 33(4): 261280.
Koerper, G.J., Richardson C.J., 1980. Biomass and net annual primary production regressions for Populus grandientata on three sites in northern lower Michigan. Can. J. For. Res. 10: 92101.
Lemke 1973. Charakterystyka ilo¶ciowa igliwia i ulistnionych gał±zek w młodszych drzewostanach sosnowych. [Quantitative characterisation of needles and twigs with needles in younger pine stands]. Fol.For.Pol. Ser.A, vol. 21: 173191 [in Polish].
Lemke 1975a. Mi±ższo¶ć gałęzi drzew w drzewostanach sosnowych [Tree branch volume in the Scots pine forests]. PTPN. 1975. t. XL: 2936 [in Polish]
Lemke J. 1975b. Szacowanie ciężaru ¶wieżego igliwia sosny zwyczajnej. [Estimation the fresh needle weight of Scots pine]. Sylwan 6: 3744 [in Polish]
Lemke J. 1983. Tabele do szacowania ciężaru igliwia i uiglonych gał±zek sosny zwyczajnej. [Tables for estimation of the weight of needles and twigs with needles of the Scots pine]. Sylwan 2: 2130 [in Polish].
Lemke J. 1978. Charakterystyka ilo¶ciowa igliwia i ulistnionych gał±zek w starszych drzewostanach sosnowych. [Quantitative characteristics of the needles and twigs with needles in old Scots pine stands]. Folia For. Pol. Ser. A, vol. 23: 5366.
Lemke J. WoĽniak A. 1992. Szacowanie masy igieł nasłonecznionej i ocienionej czę¶ci korony sosny zwyczajnej. [Estimating Needle Biomass from Sunexposed and Shaded of Scots Pine Crown]. Sylwan 2:2532.
Mäkelä A., Vanninen P. 1998. Impact size and competition on tree form and distribution of aboveground biomass in Scots Pine. Can. J. For. Res. 28: 216227.
Oleksyn J., Reich P.B., Chałupka W., Tjoelker M.G. 1999. Differential above and belove –ground biomass accumulation of European Pinus sylvestris populations in a 12year –old provenance experiment. Scand. J. For. Res. 14: 717.
Sit V., PoulinCostello M. 1994. Catalogue of curves for curve fitting. Biometrics information handbook series, no. 4. Ministry of Forests Province of British Columbia.
Vanninen P., Ylitalo H., Sievänen R. Mäkaelä A., 1996. Effects of age and site quality on the distribution of biomass in Scots pine (Pinus sylvestris L.). Trees 10, 231238.
Wagenführ R., Scheiber Chr. 1985. Holzatlas, VEB Fachbuchverlag Leipzig 2. Auflage.
Wężyk P. Kozioł K., Madejczyk A., 2001. Zakładanie sieci powierzchni monitoringowych w terenach le¶nych metod± DGPS. [Localization of the monitoring plots network in the area of woodlands using the DGPS method] I Krajowa Konferencja Systemy Informacji Przestrzennej w Lasach Państwowych. Rogów. Internet: http://www.lasypanstwowe.gov .pl/sip/Aktualia/Konfrogow/Konferencja%20rogow.htm [in Polish].
Jarosław Socha
Department of Forest Mensuration
Agricultural University of Cracow
Al. 29 Listopada 46, 31425 Cracow, Poland
email: rlsocha@cyfkr.edu.pl
Piotr Wężyk
Department of Forest Ecology
Agricultural University of Cracow
Al. 29 Listopada 46, 31425 Cracow, Poland
email: rlwezyk@cyfkr.edu.pl
Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed ‘Discussions’ in each series and hyperlinked to the article.