## 2 Methods
### Statistical Analyses
#### 2.4.1 Detecting Trends in the Fine-Root Phenology Using Generalized Additive Mixed Models  
Due to the relatively long period between sampling sessions, we seldom accurately observed elongation within one single root. Therefore, only two variables could be used to present the trend in the fine-root phenology per species: the new root count per tube (i.e., the root initiation rate or difference in the number of new roots between two consecutive sessions) and the cumulative root surface per tube (i.e., the sum of each roots' surface area per tube; in mm\({}^{2}\)). As we did not account for the root ontology or order, each root count here actually reflects an observed root tip. Each root was classified by its diameter d (d! 2 mm, 1 mm! d! < 2 mm, 0.5 mm! d! < 1 mm, and d! 0.5 mm) and season of birth (spring, summer, autumn, and winter). Unless otherwise specified, the uncertainty presented here is the variability among the twelve to twenty-four tubes per species. The trend in mean new root count and cumulative root surface was assessed using generalized additive models (GAMs) built using the _gam_ function in R/mgcv [111; 112]. Although GAMs are not as flexible as generalized additive models for location, scale, and shape (GAMLSS), we used GAMs here to ease the plotting, diagnostic testing, and interpretation of the model results using the _summary_, _gam_,_check_, _acf_, _pac_, and _plot_ functionin R/mgcv and the _confint_ function in R/gratia [113; 114; 115]. The downside of GAMs in comparison to more elaborate models like GAMLSS is that information on the trend in the variance, skewness, and kurtosis is lost [116]. Therefore, we comment below on potential model improvements where interesting. To model the new root count and the cumulative root surface as functions of their covariates, two negative binomial GAMs with the default logarithmic link function and two GAMs with the gamma distribution and the default inverse link functions were made, respectively. The GAMs with the negative binomial and gamma distribution families were chosen because they showed the lowest Akaike Information Criterion (AIC). Two GAMs (one with a negative binomial and one with a gamma distribution) were used to model the cumulative root surface and new root count of the oak trees in the PB and the birch and beech trees in the KS (Models 1-2). The fixed covariates of these GAMs were the _day_ (counted from the first of January; continuous) and _species_ (categorical with three levels). These two covariates were modeled as a factor-smooth interaction term and smoothed using P-splines [117; 118]. Due to the high number of zeros in the data, two other GAMs (again one with a negative binomial and one with a gamma distribution) were used to model the cumulative root surface and the new root count of the beech trees in the PB (Models 3-4). The fixed covariate of these GAMs was the _day_, which was also smoothed using P-splines. In all GAMs, the _individual tree_ (categorical with 16 levels) was modeled as a random intercept to account for the dependency among observations of the same individual tree. The restricted maximum likelihood (REML) argument was specified as the smoothness selection method to avoid overfitting [112; 119]. \[ \begin{split}\mathrm{Y}_{ij}\sim\mathrm{NB}(\mathrm{u}_{ij},\ \mathrm{k})\\ \mathrm{g}(\mathbb{E}(\mathrm{Y}_{\bar{y}}))=\mathrm{g}(\mathrm{u}_{ ij})\\ \mathrm{g}(\mathrm{u}_{\bar{y}})=\mathrm{Species}_{\bar{y}+}f( \mathrm{Day}_{\bar{y}},\mathrm{Species}_{\bar{y}})+\mathrm{Individual\ tree}_{i} \end{split}