## Abstract

### Background

Generalized height-diameter curves based on a re-parameterized version of the Korf function for Norway spruce (*Picea abies* (L.) Karst.), Scots pine (*Pinus sylvestris* L.) and silver birch (*Betula pendula* Roth) in Norway are presented. The Norwegian National Forest Inventory (NFI) is used as data base for estimating the model parameters. The derived models are developed to enable spatially explicit and site sensitive tree height imputation in forest inventories as well as future tree height predictions in growth and yield scenario simulations.

### Methods

Generalized additive mixed models (gamm) are employed to detect and quantify potentially non-linear effects of predictor variables. In doing so the quadratic mean diameter serves as longitudinal covariate since stand age, as measured in the NFI, shows only a weak correlation with a stands developmental status in Norwegian forests. Additionally the models can be locally calibrated by predicting random effects if measured height-diameter pairs are available. Based on the model selection of non-constraint models, shape constraint additive models (scam) were fit to incorporate expert knowledge and intrinsic relationships by enforcing certain effect patterns like monotonicity.

### Results

Model comparisons demonstrate that the shape constraints lead to only marginal differences in statistical characteristics but ensure reasonable model predictions. Under constant constraints the developed models predict increasing tree heights with decreasing altitude, increasing soil depth and increasing competition pressure of a tree. A two-dimensional spatially structured effect of UTM-coordinates accounts for the potential effects of large scale spatially correlated covariates, which were not at our disposal. The main result of modelling the spatially structured effect is lower tree height prediction for coastal sites and with increasing latitude. The quadratic mean diameter affects both the level and the slope of the height-diameter curve and both effects are positive.

### Conclusions

In this investigation it is assumed that model effects in additive modelling of height-diameter curves which are unfeasible and too wiggly from an expert point of view are a result of quantitatively or qualitatively limited data bases. However, this problem can be regarded not to be specific to our investigation but more general since growth and yield data that are balanced over the whole data range with respect to all combinations of predictor variables are exceptional cases. Hence, *scam* may provide methodological improvements in several applications by combining the flexibility of additive models with expert knowledge.