Key message Fitting and comparing three sets of compatible biomass models for prediction of biomass or carbon stocks of natural and planted Quercus variabilis Blume forests.Abstract To make the sum of estimated values from biomass models of various components of a tree equal to estimated tree total biomass for Quercus variabilis Blume (cork oak) forests in North China, single-tree compatible biomass models were developed. 100 trees from 100 plots in North China were felled to obtain biomass of aboveground components, and roots of 19 of those trees were extracted for measurement of root biomass. After Box-Cox transformations of variables, two sets of independent component biomass models with a dummy variable in models to define stand origin were separately built using linear mixed effects analyses (one set of models with site as a random factor; the other set without any random factor). Then three methods were compared to force compatibility of the biomass models: sums of linear mixed effects models, sums of linear models, and simultaneous equation fits based on linear models. Model parameters were estimated by ordinary least squares (OLS) or seeming unrelated regression procedures (SUR). Coefficients of determination (r 2 ), root mean square error (Rmse), residuals plots and histograms of residuals indicated that models fitted with sums of linear mixed effects models were the least biased and most precise at estimating total above-ground biomass.Further testing for the linear mixed effects models with jackknife validation and prediction sum of squares (PRESS) statistics indicated that the compatible biomass models can be used to predict biomass or carbon stocks of cork oak forests in North China within specific tree diameter at breast height and height ranges.

Compatible taper and volume models were created for Quercus variabilis Blume (cork oak) forests in North China. 174 trees were felled to obtain stem analysis data. Linear mixed effects analyses were used in modelling. Firstly, a bark thickness model was built. Then diameter at breast height over bark (DBHob) for the inner layers of the 174 trees could be calculated, based on which a total volume model was built. The estimated volume and a specific parameter restriction were then substituted into a polynomial taper model, finally the taper model was fitted and compatible taper and volume models were obtained. Four sets of models based on different data sets were separately built and compared through coefficients of determination (R 2 ), root mean square error (RMSE), value of Akaike's information criterion (AIC), residuals plots and histograms of residuals. Models based on data of the analyzed stems without ramicorns and simultaneously with relative diameter under 1.5 were chosen as the most precise. Further testing of the chosen models using the jackknife method for the bark thickness and total volume models and a validation data set for the taper model verified that those models can be used to predict bark thickness, diameter at a specific point along the stem, merchantable volume and total stem volume of cork oak forests in North China within specific tree diameter at breast height and height ranges.