## Materials and Methods
### Biometric Measurements and Estimation of NPP\({}_{bm}\)
#### 2.5.2 Modelling Seasonal Dynamics of \(NPP_{bm}\)  
The stem increment of deciduous trees growing in the continental parts of Europe starts in the spring before the new leaves develop. Part of that stem increment is due to tree re-hydration and it should not be considered as growth in terms of increase in carbon stock. Furthermore, although all of the carbon in leaf biomass is usually accounted as part of the current season's \(NPP\), a part of that carbon actually comes from tree reserves of the previous season(s) and not from carbon that is fixed in the current season [28]. On the other hand, stem increment almost ceases in summer, particularly after the mid of August [6]. However, from the CO\({}_{2}\) flux measurements we can observe that the forest continues to accumulate carbon; apparently more than the amount required for the growth of stems and branches (see Section 3). The fixed carbon is likely stored in tree reserves [6]. These facts should be addressed when comparing seasonal dynamics of \(NPP\) estimates from eddy covariance and biometric measurements. Since the measurement and assessment of changes in trees' reserves is very challenging, and it was beyond the scope of this paper, in order to account for the seasonal \(NPP_{LF}\) we assumed a simplistic model where the amount of carbon mobilised from trees' reserves and used in early spring for the formation of new tissues (primarily that of leaves), is approximately equal to the \(NPP_{LF}\). Consequently, the \(NPP_{LF}\) of the current season should not be accounted for in the spring, but only later in the season, when the reserves are being replenished. The underlying idea is to account, in a simple and transparent way, for the effects of NSC storage changes. The actual mechanism of carbon reserve management in trees is significantly more complex and is currently still being researched [6; 6; 16; 23; 25; 67; 68; 69]. In our simple model, we assumed that the production of every tree component under the average growing conditions exhibits the pattern, which can be described by the logistic curve of the form:  
\[NPP_{i}(DOY)=\frac{NPP_{i,annual}}{1+e^{-k(DOY-DOY_{max}(i))}} \tag{10}\]  
where \(NPP_{i}(DOY)\) is the cumulative net primary production of the \(i\)-\(th\) component (e.g., stem and branches, leaf, fruit) from the first until a given Day-Of-Year (\(DOY\)), \(NPP_{i-annual}\) is the annual \(NPP\) of the component \(i\), \(DOY_{max}(i)\) is the \(DOY\) when the rate-of-change of \(NPP(i,\,DOY)\) is at its maximum (i.e., the inflexion point of the annual \(NPP\) curve), and \(k\) is a parameter that is related to the maximum rate-of-change of \(NPP\) (\(r_{max}\)) where \(r_{max}\) is the quotient of the maximum daily net primary production (\(dNPP_{max}\)) and the annual net primary production (\(NPP_{annual}\)):  
\[r_{max}=\frac{dNPP_{max}}{NPP_{annual}}. \tag{11}\]  
If \(r_{max}\) is known (or estimated) the parameter \(k\) can be calculated using the following equation:  
\[k=4\times atanh(r_{max})=2\times ln\left(\frac{1+r_{max}}{1-r_{max}}\right) \tag{12}\]  
Inversely,  
\[r_{max}=tanh\left(\frac{k}{4}\right)=\frac{e^{k/4}-e^{-k/4}}{e^{k/4}+e^{-k/4}} \tag{13}\]  
Using the nl routine for nonlinear least-squares estimation in STATA 14 statistical software (StataCorp LP, College Station, TX, USA), we fitted the \(NPP_{WBI}\) with the Equation (10) for each year separately and obtained estimates for the parameters \(NPP_{WBI}\),\(annual\), \(DOY_{max\_WBI}\) and \(k\) (Table 10). Seasonal dynamics of \(NPP_{LF}\) was estimated assuming that the shape parameter \(k\) for the \(NPP_{LF}\) is the same as for \(NPP_{WBI}\). Parameter \(NPP_{LF\_annual}\) equals the measured \(NPP_{LF}\).