Tree diameter distribution models represent useful tools for predicting stand growth under various environmental conditions. In this context, the Weibull probability density function is widely used in forestry due to its flexibility. In this study, this probability density function was applied to analyse the pedunculate oak Quercus robur stand structure in different forest types. The Weibull function demonstrated a high ability to fit well the empirical distribution of oak tree diameters in all studied stands. The analysis revealed either moderate variability or relative homogeneity in the stand structure, depending on the forest type. In the pedunculate oak forests with cherry Prunus avium in the northern region and those with blackthorn P. spinosa in the southern region, developed on dry soils, the parameters b and c of the Weibull function indicated moderate variability in stem diameters. In contrast, the pedunculate oak forests with hornbeam Carpinus betulus, developed on fresh soils, exhibited a more homogeneous structure. To evaluate the relationship between the b and c parameters of the Weibull function and dendrometric indices (AMD, Hmax, QMD, and Dmax), reciprocal−y square root−x and multiplicative regression models were used. The results emphasized the superiority of the multiplicative model in describing the relationships between the c parameter and dendrometric indices, compared to the reciprocal−y square root−x model, which showed modest performance for the b parameter, according to the R² coefficient.

Aranda, G., 1990. Los bosques flotantes: historia de un roble del siglo XVIII. Madrid: ICONA, 231 pp. Bailey, R.L., Dell, T.R., 1973. Quantifying diameter distributions with the Weibull function. Forest Science, 19 (2): 97-104. DOI: https://doi.org/10.1093/forestscience/19.2.97. Bassil, S., Nyland, R.D., Kern, C.C., Kenefic, L.S., 2019. Dynamics of the diameter distribution after selection cutting in uneven- and even-aged northern hardwood stands: A long-term evaluation. Canadian Journal of Forest Research, 49 (12): 1525-1539. DOI: https://doi.org/10.1139/cjfr-2019-020. Bauhus, J., Forrester, D.I., Pretzsch, H., Felton, A., Pyttel, P., Benneter, A., 2017. Silvicultural options for mixed-species stands. In: J. Bauhus, D.I. Forrester, H., Pretzsch, eds. Mixed-species forests. Berlin/Heidelberg: Springer, pp. 433-501. Bessonova, V.P., Yakovlieva-Nosar, S.O., 2024. Peculiarities of water exchange of Quercus robur and Acer campestre in an oak-field maple forest. Regulatory Mechanisms in Biosystems, 15 (3): 490-495. DOI: https://doi.org/10.15421/022469. Bose, A.K., Scherrer, D., Camarero, J.J., Ziche, D., Babst, F., Bigler, C., Bolte, A., Dorado-Lińán, I., Etzold, S., Fonti, P., Forrester, D.I., Gavinet, J., Gazol, A., de Andrés, E.G., Karger, D.N., Lebourgeois, F., Lévesque, M., Martínez-Sancho, E., Menzel, A., Neuwirth, B., Nicolas, M., Sanders, T.G., Scharnweber, T., Schröder, J., Zweifel, R., Gessler, A., Rigling, A., 2021. Climate sensitivity and drought seasonality determine post-drought growth recovery of Quercus petraea and Quercus robur in Europe. Science of the Total Environment, 784: 147222. DOI: https://doi.org/10.1016/j.scitotenv.2021.147222. Brosofske, K.D., Froese, R.E., Falkowski, M.J., Banskota, A., 2014. A review of methods for mapping and prediction of inventory attributes for operational forest management. Forest Science, 60 (4): 733-756. DOI: http://dx.doi.org/10.5849/forsci.12-134. Ciceu, A., Pitar, D., Badea, O., 2021. Modeling the diameter distribution of mixed uneven-aged stands in the South Western Carpathians in Romania. Forests, 12 (7): 958. DOI: https://doi.org/10.3390/f12070958. Cuza, P., 2017. Specificul distribuţiei descendenţilor după înălţime în culturile experimentale ale stejarului pedunculat (Quercus robur). [The specific distribution of descendants after height in experimental crops of pedunculate oak (Quercus robur)]. Studia Universitatis Moldaviae, Seria A, Ştiinţe reale şi ale naturii, 6 (106): 35-40. Cuza, P., 2019. Distribuţia puieţilor stejarului pedunculat (Quercus robur L.) pe categorii de diametre în culturile de descendenţe materne. [The distribution of pedunculate oak saplings (Quercus robur L.) by categories of diameters in maternal offspring cultures]. Revista Botanică, 11 (2): 18-27. Dervishi, V., Poschenrieder, W., Rötzer, T., Moser-Reischl, A., Pretzsch, H., 2022. Effects of climate and drought on stem diameter growth of urban tree species. Forests, 13: 641. DOI: https://doi.org/10.3390/f13050641. Diamantopoulou, M.J., Özçelik, R., Crecente-Campo, F., Eler, Ü., 2015. Estimation of Weibull function parameters for modelling tree diameter distribution using least squares and artificial neural networks methods. Biosystems Engineering, 133: 33-45. DOI: https://doi.org/10.1016/j.biosystemseng.2015.02.013. Durbin, J., Watson, G.S., 1950. Testing for serial correlation in least squares regression. Biometrika, 37 (3-4): 409-438. Egonmwan, I.Y., Ogana, F.N., 2020. Application of diameter distribution model for volume estimation in Tectona grandis L.f. stands in the Oluwa forest reserve, Nigeria. Tropical Plant Research, 7: 573-580. DOI: https://doi.org/10.22271/tpr.2020.v7.i3.070. Fonseca, T.F., Marques, C.P., Parresol, B.R., 2009. Describing maritime pine diameter distributions with Johnson’s SB distribution using a new all-parameter recovery approach. Forest Science, 55: 367-373. DOI: https://doi.org/10.1093/forestscience/55.4.367. Fortin, M., Van Couwenberghe, R., Perez, V., Piedallu, C., 2019. Evidence of climate effects on the height-diameter relationships of tree species. Annals of Forest Science, 76: 1-20. DOI: https://doi.org/10.1007/s13595-018-0784-9. Fu, Y., He, H.S., Wang, S., Wang, L., 2022. Combining Weibull distribution and k-nearest neighbor imputation method to predict wall-to-wall tree lists for the entire forest region of Northeast China. Annals of Forest Science, 79 (1): 42. DOI: https://doi.org/10.1186/s13595-022-01161-9. Gabdelkhakov, A., Rakhmatullin, Z., Martynova, M., Fazlutdinov, I., Mullagaleev, I, 2021. Evaluating diameter distribution series of small-leaved lime (Tilia cordata Mill.) in forest stands. Plant Methods, 17: 45. DOI: https://doi.org/10.1186/s13007-021-00741-6. Gheideman, Ň.C., Ostapenko, B.F., Nicolaeva, L.P. et al., 1964. Forest types and forest associations in the Moldavian SSR. Chisinau: Moldovan book, 268 pp. Gorgoso-Varela, J.J., Adedapo, S.M., Ogana, F.N., 2020. A comparison between derivative and numerical optimization methods used for diameter distribution estimation. Scandinavian Journal of Forest Research, 35: 156-164. DOI: https://doi.org/10.1080/02827581.2020.1760343. Gorgoso-Varela, J.J., Rojo, A., Camara-Obregon, A., Dieguez-Aranda, U., 2012. A comparison of estimation methods for fitting Weibull, Johnson’s SB and beta functions to Pinus pinaster, Pinus radiate and Pinus sylvestris stands in northwest Spain. Forest Systems, 21 (3): 446-459. DOI: https://doi.org/10.5424/fs/2012213-02736. Gorgoso-Varela, J.J., Rojo-Alboreca, A., 2014. Use of Gumbel and Weibull functions to model extreme values of diameter distributions in forest stands. Annals of Forest Science 71: 741-750. DOI: https://doi.org/10.1007/s13595-014-0369-1. Guo, H., Liu, X., Liu, D., 2024. Soil-sensitive Weibull distribution models of Larix principis-rupprechtii plantations across northern China. Forests, 15 (9): 1562. DOI: https://doi.org/10.3390/f15091562. Kint, V., Leuven, K., Robert, D.W., Noël, L., 2004. Evaluation of sampling methods for the estimation of structural indices in forest stands. Ecological Modelling, 180 (4): 461-476. DOI: https://doi.org/10.1016/j.ecolmodel.2004.04.032. Litovchenko, D.A., Popova, A.A., Shestibratov, K.A., Krutovsky, K.V., 2024. Climate response and radial growth dynamics of pedunculate oak (Quercus robur L.) plus trees and their half-sib progeny in periods of severe droughts in the forest-steppe zone of eastern Europe. Plants, 13 (22): 3213. DOI: https://doi.org/10.3390/plants13223213. Liu, C., Beaulieu, J., Pregent, G., Zhang, S., 2009. Applications and comparison of six methods for predicting parameters of the Weibull function in unthinned Picea glauca plantations. Scandinavian Journal of Forest Research, 24 (1): 67-75. Martins, W.B.R., Douglas Roque Lima, M., De Oliveira Barros Junior, U., Sousa Villas-Boas Amorim, L., De Assis Oliveira, F., Schwartz, G., 2020. Ecological methods and indicators for recovering and monitoring ecosystems after mining: A global literature review. Ecological Engineering, 145: 105707. DOI: https://doi.org/10.1016/j.ecoleng.2019.105707. Mauro, F., Frank, B., Monleon, V.J., Temesgen, H., Ford, K.R., 2019. Prediction of diameter distributions and tree-lists in southwestern Oregon using LiDAR and stand-level auxiliary information. Canadian Journal of Forest Research, 49 (7): 775-787. DOI: https://doi.org/10.1139/cjfr-2018-0332. McElhinny, C., Gibbons, P., Brack, C., Bauhus, J., 2005. Forest and woodland stand structural complexity: Its definition and measurement. Forest Ecology and Management, 218 (1-3): 1-24. DOI: https://doi.org/10.1016/j.foreco.2005.08.034. Merganič, J., Sterba, H., 2006. Characterization of diameter distribution using the Weibull function: Method of moments. European Journal of Forest Research, 125 (4): 427-439. DOI: https://doi.org/10.1007/s10342-006-0138-2. Mey, R., Temperli, C., Stillhard, J., Nitzsche, J., Thürig, E., Bugmann, H., Zell, J., 2023. Deriving forest stand information from small sample plots: An evaluation of statistical methods. Forest Ecology and Management, 544: 121155. DOI: https://doi.org/10.1016/j.foreco.2023.121155. Návar, J., 2014. A stand-class growth and yield model for Mexico’s northern temperate, mixed and multiaged forests. Forests, 5: 3048-3069. DOI: https://doi.org/10.3390/f5123048. Newton, P.F., Amponsah, I.G., 2005. Evaluation of Weibull-based parameter prediction equation systems for black spruce and jack pine stand types within the context of developing structural stand density management diagrams. Canadian Journal of Forest Research, 35: 2996-3010. DOI: https://doi.org/10.1139/cjfr-2020-0289. Ogana, F.N., Osho, J.S.A., Gorgoso-Varela, J.J., 2015. Comparison of Beta, Gamma and Weibull distributions for characterising tree diameter in Oluwa Forest Reserve, Ondo State, Nigeria. Journal of Natural Sciences Research, 5 (4): 28-36. Postolache, Gh., 1995. Vegetaţia Republicii Moldova. Chişinău: Ştiinţa, 340 pp. Pretzsch, H., 1999. Modelling growth in pure and mixed stands: A historical overview. In: A.F.M. Olsthoorn, H.H. Bartelink, J.J Gardiner, H. Pretzsch, H.J. Hekhuis, A. Franc, eds. Management of mixed-species forest: Silviculture and economics. Wageningen: IBN-DLO, pp. 102-107. Pretzsch, H., Forrester, D.I., Bauhus, J., 2017. Mixed-species forests: Ecology and management. Heidelberg: Springer, 653 pp. Pretzsch, H., Schütze, G., 2016. Effect of tree species mixing on the size structure density and yield of forest stands. European Journal of Forest Research, 135 (1): 1-22. DOI: https://doi.org/10.1007/s10342-015-0913-z. Sa, Q., Jin, X., Pukkala, T., Li, F., 2023. Developing Weibull-based diameter distributions for the major coniferous species in Heilongjiang Province, China. Journal of Forestry Research, 34: 1803-1815. DOI: https://doi.org/10.1007/s11676-023-01610-9. Schmidt, L.N., Sanquetta, M.N.I, McTague, J.P., da Silva, G.F., Fraga Filho, C.V., Sanquetta, C.R., Soares Scolforo, J.R., 2020. On the use of the Weibull distribution in modeling and describing diameter distributions of clonal eucalypt stands. Canadian Journal of Forest Research, 50 (10): 1050-1063. DOI: https://doi.org/10.1139/cjfr-2020-0051. Schütz, J.P., Rosset, C., 2020. Performances of different methods of estimating the diameter distribution based on simple stand structure variables in monospecific regular temperate European forests. Annals of Forest Science, 77: 1-11. DOI: https://doi.org/10.1007/s13595-020-00951-3. Shiver, B.D., 1988. Sample sizes and estimation methods for the Weibull distribution for unthinned slash pine plantation diameter distributions. Forest Science, 34 (3): 809-814. DOI: https://doi.org/10.1093/forestscience/34.3.809. Sohrabi, H., Taheri Sarteshnizi, M.J., 2013. Probability distribution functions for modeling diameter distribution of oak species in pollarded northern Zagros forests. Iranian Journal of Forest, 4: 333-343. Vospernik, S., Heym, M., Pretzsch, H., Pach, M., 2023. Tree species growth response to climate in mixtures of Quercus robur/Quercus petraea and Pinus sylvestris across Europe – a dynamic, sensitive equilibrium. Forest Ecology and Management, 530 (2): 120753. DOI: https://doi.org/10.1016/j.foreco.2022.120753.