Digital Appendix

Digital Appendix

Quantifying general growth trends

The division into two periods (before 1990 and past 1990) was made due to increased urban development measures starting in Berlin after 1990.

The idea of the analysis was to fit a biologically plausible and thus extrapolatable growth equation to each tree’s basal area development in the period up to 1990. Comparing the extrapolated model’s predictions with the real basal area development after 1990 should reveal growth trends in the second period. We chose the Bertalanffy growth function (Zeide 1993), but with an additional additive parameter () which covers the uncertainty of tree age estimation.

(1)

with

:estimated basal area (cm²/yr)

t: age

a, b:parameters of the original Bertalanffy equation

c: additive parameter as explained in the text.

The parameter estimates for each tree were obtained by nonlinear least squares regression.

The annual deviations () of a tree’s actual basal area () from the extrapolated Bertalanffy curve values () obtained from equation 1 were calculated as

(2)

with the indices and referring to a given tree and a given calendar year, respectively.

For further use the subsequent calculation of the basal area increment (ig) was derived by building the difference of the recent value with the previous year value:

(3)

with indicating the current calendar year.

For analyzing the growth differences between the two observation periods a ratio of the measured values were calculated as follows:

(4)

with : basal increment from the measured values.

Sensitivity of the annual tree increment

In our case a comparison of two observation periods is derived due to obvious differences in climate with the aim to analyze the growth patterns within these two periods. For this we used a linear mixed effects model of the following form:

(5)

with

: Variance of the ring width index

: indexes a given tree

: indicating the period up to 1990 (j = 1) and after 1990 (j = 2).

: Dummy variable for housing density (MD = 1 for medium housing density, MH = 0 otherwise)

: Dummy variable for housing density (HH = 1 for high housing density, HH = 0 otherwise)

: Dummy variable for period of interest (PER = 1 for the time before 1990, PER = 0 otherwise)

: Tree age at the time of coring

: Random effect on tree level,

: i.i.d. errors.

The tree-level random effect accounts for the two values of VRWI per tree (one for the first, one for the second period) which cannot be assumed to be uncorrelated.

Analysis of the relationship between growth and climate parameters

We fitted a mixed linear model for a dataset including growth and climate data of the high and low-dense trees. Thus we analyzed the relation between climate parameters and tree growth. The following equation describes the statistical model:

(6)

with

: indexes a given tree

: indicating the period up to 1990 (j = 2) and after 1990 (j = 1).

: Ring width index

: Annual air temperature (°C)

: Annual precipitation sum (mm)

: Dummy variable for period of interest (PER = 1 for the time after 1990, PER = 0 otherwise)

HH: Housing density as defined above

: Random effect on tree level,

: i.i.d. errors.

Based on this model linear combinations of the estimated model parameters , , and were tested for significant effects on tree growth, represented by . In a second estimation, the dummy variable gv (glacial valley) was added to the above shown model (Eq. 6). As the variable did not show a significant effect, the results are not shown in this article.