## Abstract

Since biomass is one of the key variables in ecosystem studies, widespread effort has aimed to facilitating its estimation. Numerous stand-specific volume and biomass equations are available, but these cannot be used for scaling up biomass to the regional level where several age-classes and structural types of stands coexist. Therefore simplified generalized volume and biomass equations are needed. In the present study, generalized biomass and volume regression equations were developed for the main tree species in Europe. These equations were based on data compiled from several published studies and are syntheses of the published equations. The results show that these generalized equations explain 64–99% of the variation in values predicted by the original published equations, with higher values for stem than for crown components.

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## Notes

Forest inventories use as volume functions mostly equations other than allometric equations.

The

*R*^{2}and the root-mean-square error of the generalized volume and biomass equations were used to assess the variability of these equations relative to the original equations. These apply to the pseudoobservations generated and do not express the accuracy relative to the field data but indicate the amount of variation in predictions by the original equations accounted for by the generalized equations.

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## Acknowledgments

This study was a part of the EU-funded research consortium ‘Multi-source inventory methods for quantifying carbon stocks and stock changes in European forests’ (CarboInvent EKV2-2001-00280). I thank Dr. Raisa Mäkipää, Dr. Aleksi Lehtonen and Mr. Mikko Peltoniemi for their comments on the manuscript, Dr. Emil Cienciala for providing the aggregated height values per age-class based on data of forest management plans in the Czech Republic, Dr. James Thompson for editing the English language and the Finnish National Forest Inventory for providing data on permanent sample plots.

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## Additional information

Communicated by Michael Köhl.

## Appendices

### Appendix 1

The modelled relationships between *dbh* and *h* were tested, reserving a portion of the available data to obtain an independent measure of the model prediction accuracy. When no test set is available for model validation, a cross-validation criterion can be used (Stone 1974; Snee 1977). Model validation was accomplished with the leave-one-out (LOO) cross-validation. The dataset is split into a training set, on which a model is estimated, and a test set on which the model is evaluated. In this case the response value *ŷ*
_{(i)} is predicted on a model that was estimated for the dataset minus the *i*th observation, while the test set contains only one observation (Stone 1974). The splitting procedure is repeated until all observations have once and only once been in the test set. Thus there are *n* models built, each using *n−*1 observations for model construction and the remaining observations for model validation. The LOO cross-validation criterion mPRESS (mean of the predictive error sum of squares) is most often used (Stone 1974; Snee 1977):

in which *n* is the number of observations in the test set and *y*
_{i} and *ŷ*
_{(i)} are, respectively, the experimental and predicted response values. Taking the square root of this, we can derive the root-mean-square error of cross-validation:

The relative mean error of cross-validation was also calculated:

### Appendix 2

The dissimilarity \( {\left( {H_{0} :\beta ^{{{\text{BOR}}}}_{1} = \beta ^{{{\text{TEM}}}}_{1} ;H_{1} :\beta ^{{{\text{BOR}}}}_{1} \ne \beta ^{{{\text{TEM}}}}_{1} } \right)} \) of parameters \( \beta ^{{{\text{BOR}}}}_{1} \) and \( \beta ^{{{\text{TEM}}}}_{1} \) (Table 3) was tested using the test statistic t (Ranta et al. 1999):

where the standard error \( s_{{\beta ^{{{\text{BOR}}}}_{1} - \beta ^{{{\text{TEM}}}}_{1} }} \)is given by

where the *n*
_{BOR} and the *n*
_{TEM} are the sample sizes. The error variance was assumed to be of equal size in both populations. The
\( {\left( {s_{{Y \cdot X}} ^{2} } \right)}_{p} \) is the so-called pooled variance estimator and is given by

The \( SS_{{{\text{RESIDUAL}}}} ^{j} \)follows:

The \( {\left( {n^{j} - 1} \right)}s_{X} ^{2} {\left( j \right)} \), \( {\left( {n^{j} - 1} \right)}s_{Y} ^{2} {\left( j \right)} \) and \( {\left( {n^{j} - 1} \right)}s_{{XY}} {\left( j \right)} \) were calculated from

and

respectively.

If the error variance is normally distributed and the errors of different values of the explanatory variable are independent, the test statistic follows the *t*-distribution with
\( n_{{{\text{BOR}}}} + n_{{{\text{TEM}}}} - 4 \) degrees of freedom.

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Muukkonen, P. Generalized allometric volume and biomass equations for some tree species in Europe.
*Eur J Forest Res* **126**, 157–166 (2007). https://doi.org/10.1007/s10342-007-0168-4

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DOI: https://doi.org/10.1007/s10342-007-0168-4