Spatial distribution of microbial biomass and respiration (H1)
To test the effects of distance to the closest tree and depth on the soil microbial biomass and respiration, we used linear mixed-effects models and normal distribution assumptions that included plot as a random effect, and distance and depth as fixed effects. The model was fitted on monospecific pairs and was used to predict the soil microbial properties over a distance to the closest tree from 0 to 90 cm and a depth from 0 to 10 cm (Suppl. S2).
Belowground overyielding between heterospecific tree pairs (H2) Belowground overyielding of soil microbial biomass and respiration was calculated as the difference between observed soil microbial properties between a heterospecific pair and what would be expected based on the weighted means of the monospecific pairs for a given position between the trees (overyielding (position = i, depth = j) = observed (i, j) – ((expectedL. formosana (i, j) + expected S. saponaria (i, j)) / 2)), where ”i ” is the position between the trees and ”j ” is the depth. Positive results indicate soil microbial properties in mixed pairs are overyielding (i.e., producing more biomass or respiration than expected based on monospecific pairs), and negative results indicate soil microbial properties in mixed pairs are underyielding (i.e., producing less biomass or respiration than expected based on monospecific pairs). The expected values were predicted from the model fits from H1. We used belowground overyielding as a response variable to test for the effect of the heterospecific pair on the aggregated soil samples, as well as for the effect of depth using a linear mixed-effects model with plot as random effect and pair as fixed effect.    To determine differences between soil depths, we used a Tukey HSD test based on an analysis of variance (ANOVA type 1).
Spatial distribution of belowground overyielding (H3) To test the effects of distance to the tree species and depth on the belowground overyielding of soil microbial biomass and respiration, we used linear mixed-effects models, which included plot as a random effect and distance in centimeters from the trees and depth as fixed effects (Suppl. S2). We fixed the positions of the trees to L. formosanabeing tree 1 and S. saponaria being tree 2 in a mixed pair. Like this, the positioning of the trees was fixed within the data and could be analysed in terms of a spatial gradient.
All linear mixed-effect models were fitted, using the ”lmer” function of the R package lme4 (Bates et al. 2015). To define the quality of the model fits of all used linear mixed-effects models, the ”check_model” function of the R package performance (Lüdecke et al. 2021) was used to investigate various model assumptions, such as normality of residuals, normality of random effects, linear relationship, homogeneity of variance, and multicollinearity (Briggs and Cheek 1986).