140 Lukaszkiewicz and Kosmala: Determining the Age of Streetside Trees by substituting a particular tree age (10, 15, 20 years, and so on) into the regression model under research and then by introducing the dbh value typical for the particular age. With known values of age and dbh parameters, the third value of total height was determined by an iteration approach. Iteration means matching the third parameter to the values of the other two by multiple subsequent approximations (Motulsky and Christopoulos 2003). This article used GraphPad application (GraphPad Software, Inc., San Diego, CA) for the purpose. In this way, we obtained a number of pairs of dendrometric parameters for a known age, that is, the dbh and total height which, after being substituted into the model concerned and calculated, determine the particu- lar age. Consecutive sets of iterations allowed obtaining a number of Figure 4. Horsechestnut (Aesculus hippocastanum L.) nomo- gram. The nomogram contains age lines in 5-year intervals in a range of 10 to 60 years. The area limited with hashed lines contains 95% calculated age values of all unrelated trees of known age taken from a control group. coefficient (r) was presented together with its significance level (P) and determination coefficient (r2). Regression and determi- nation coefficients elaborated for each species indicated that over 90% of age value variation is explained by dbh and tree height values variation (Table 2). RESULTS Determining of Multifactorial Regression Model The obtained nomograms (Figures 2, 3, and 4) are an attempt to transfer a tridimensional interrelation (three parameters: age, dbh, height) onto a flat surface, which by nature requires using a two-coordinate system. For this reason, it is impossible to read tree age as a continuous parameter and an approximate readout is enforced, e.g., in 5-year intervals, as used in this article. The lines separating age brackets on the nomograms were obtained Table 2. Equation factors of regression model for each tree species in which Age – the calculated age of street tree population (years); DBH – mean dbh (cm) taken from street trees population; H – mean total height (m) of street tree population. Factor a b c d r P r2 Common lime 264.073 5.5834 0.3397 0.0026 0.9491 <0.001 0.9009 dbh diameter at breast height. ©2008 International Society of Arboriculture Common ash 210.115 5.3523 0.2655 0.0064 0.9857 <0.001 0.9716 Horsechestnut 54.2714 4.0709 0.7988 0.0209 0.9635 <0.001 0.9284 points for the adopted age value, making possible to draw a line reflecting the specific age value (e.g., 10, 15, 20 years, and so on). The calculations were performed for each age line sepa- rately. For instance, a nomogram developed for common lime (Figure 1) shows 16 age lines marking 5-year brackets between the ages of 10 and 85 years. Afterward, age division lines were marked on the coordinate system in which the X-axis reflected dbh values and the Y-axis reflected total tree height. The area between thick broken lines contains 95% of tree age values from the control group. The lines were drawn based on prediction intervals for the dbh/height relation. Application of Nomograms Subsequently is a sample method of a potential field application of a chart with a nomogram. Reading the age with the use of the nomogram should be done especially for these trees complying with the conditions set up in this work (e.g., six streetside com- mon limes). Measurement of dbh and height (H) for each tree is taken. Then mean dbh and mean height for all measured trees in this population is calculated with the result that, for six investi- gated common lime trees, mean dbh values is 50 cm (20 in) and the mean height values are 16 m (52.8 ft). Next we find the values of the mean dbh and the mean H of trees on the corre- sponding axis at the common lime nomogram and draw perpen- dicular lines. The crossing point for both dendrometric param- eters (marked with arrows) in this example is pointed at the “65 years” line (Figure 1). Therefore, the mean age of six streetside common limes is 65 years (range of error is described in the verification part). If the crossing point falls between the age lines, the value should be interpolated. Verification of the Multifactorial Regression Model The credibility of elaborated multifactorial regression model was tested. The verification did not consist of determining whether the regression model is “true” but in seeing whether it generated a fair, verifiable hypothesis on the determination of tree age based on dendrometric parameters. Initially, we compared the age of trees as determined with the model and the actual age of trees using the control group chosen randomly from the researched tree populations of each species. The control group was not involved in developing the regression model. The dendrometric data of trees from the population were used to arrive at the age of the particular tree using the math- ematic version of a particular regression model developed for the species concerned. Hence, each individual tree used to verify the
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