The objective of this study was to develop a system of compatible equations to estimate eucalyptus ( Eucalyptus globulus Labill.) tree aboveground biomass and biomass of tree components for forest biomass prediction across regional boundaries. Data came from 441 trees sampled on several sites (99 and 14 plots in planted and coppice regenerated stands, respectively) representative of the eucalyptus expansion area in Portugal. The system of equations, simultaneously fitted using seemingly unrelated regression, was based on the allometric model for the biomass of stem wood, stem bark, leaves, and branches. Total aboveground biomass was expressed as the sum of the biomass of the respective tree components. The study allowed the following conclusions: (i) there is a significant increase in the predictive ability of the models that include height (stem components) or crown length (crown components) as an additional predictor to diameter at 1.30 m; (ii) there is a clear effect of the stage of development of the stand on tree allometry, with a decreasing pattern of the allometric constants; (iii) no effect of stand density, site index or climate on tree allometry was found; and (iv) for practical purposes, the same system of equations can be used for planted and coppice regenerated stands.

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Objective The objective of the research was to develop a generalized height-diameter model for Quercus suber L. in Portugal, which can be applied both to undebarked and debarked trees, with diameter at breast height over cork larger than 2.5 cm.• Methods A nonlinear fixed effects model (NLFEM) and a nonlinear mixed effects model (NLMEM) approaches were used. Parameters estimates were obtained using the SAS macro NLINMIX, which uses a linear approximation to the marginal likelihood function by expanding it with a first-order Taylor series on the random effects. The option of expanding on the random effects at their current empirical best linear unbiased predictors (EBLUP) was used. The fitted models were evaluated using an independent data set, together with an existing model specific for undebarked trees. To obtain subject specific predictions with the NLMEM, a conventional and an improved calibration procedures were applied, considering four different tree sub-sampling designs. Both proposed models included dominant height and stand density as covariates to explain plot variability. • Conclusions Validation indicated that, even in the situations where the NLMEM calibration is not possible, this model should be preferred. The differences between the validated models, which were more evident for young stands, were considered. No large differences in predictive accuracy were found between the calibrated NLMEM using the conventional or the improved calibration procedures, for all the considered sub-sampling designs.

The work described in this paper aimed to establish a single set of parameter values for a process-based model (3-PG), applicable to Eucalyptus globulus Labill. in Portugal. Initial testing was done with parameter values from Australia using climate, biometric, and soils data from an irrigation and fertilization trial and a spacing trial. The model provided good estimates for stem mass and basal area, poor estimates for leaf mass, and reasonable estimates for volume. The fit between simulated and observed values was then improved by tuning parameter values to produce a final set. The calibrated model was tested, and performed well, against data from permanent sample plots (PSPs) at different locations across Portugal. Volume and basal area predictions made by 3-PG for PSPs were then compared with predictions made by the empirical model in use for E. globulus plantations in Portugal. Differences were negligible. Model outputs with the Australian parameter set and the optimum set for Portugal indicated that partitioning of carbohydrates (net primary productivity) was very different in E. globulus grown in Portugal and Australia. The study has confirmed the potential of this process-based model as a practical tool to support forest management decision-making.

Modelling growth of trees or stands when age is not available is often necessary. This is the case in national forest inventories or when age is not a main determinant of growth (e.g., trees growing in uneven-aged stands). Even if age is not known, functions used to model growth should follow the required pattern, with a maximum and a slow decline after the maximum is attained. There are some empirical functions with these properties; however, direct modelling with difference equations derived from the so-called theoretical growth functions has not been used for this purpose, as they are age dependent. This paper presents a methodology to formulate growth functions as ageindependent difference equations. These can be used when age is not available or is not relevant. The proposed equations have the advantage of allowing the direct modelling of yield instead of growth. If the parameters are expressed as a function of site variables, the equations are invariant for projection length and therefore can be used when data is not equally spaced in time, as is the case of most data sets. The methodology is applied to the Lundqvist and Richards growth functions, the most commonly used in growth and yield modelling. The use of the age-independent growth functions is illustrated by using two case studies in Portugal: dominant height growth of eucalyptus (Eucalyptus globulus Labill.) plantations and individual tree growth in diameter at breast height in sparse cork oak (Quercus suber L.) stands.Résumé : La modélisation de la croissance des arbres ou des peuplements est souvent nécessaire alors que l'âge n'est pas disponible. C'est le cas pour les inventaires des forêts nationales ou lorsque l'âge n'est pas le déterminant principal de la croissance, par exemple, dans le cas des arbres croissant dans des peuplements inéquiennes. Même si l'âge n'est pas connu, les fonctions utilisées pour modéliser la croissance devraient obligatoirement suivrent un patron comportant un maximum suivi d'un lent déclin une fois le maximum atteint. Il existe quelques fonctions empiriques ayant ces propriétés. Pourtant, la modélisation directe à l'aide d'équations différentielles dérivées de fonctions de croissance dites théoriques n'a jamais été utilisée à cet effet, car elles sont toutes dépendantes de l'âge. Cet article présente une métho-dologie pour formuler des fonctions de croissance sous forme d'équations différentielles indépendantes de l'âge et utilisables lorsque l'âge n'est pas disponible ou n'est pas pertinent. Les équations proposées ont l'avantage de permettre une modélisation directe du rendement au lieu de la croissance. Si les paramètres sont exprimés en fonction de variables de site, les équations sont indépendantes de l'étendue de la projection et peuvent donc être utilisées lorsque les données ne sont pas espacées régulièrement dans le temps, comme c'est le cas de la plupart des jeux de données. La méthodologie est appliquée aux fonctions de croissance de Richards et de Lundqvist qui sont les plus souvent utilisées pour modélis...