, Volume 33, Issue 1, pp 103–119 | Cite as

Modelling individual tree height–diameter relationships for multi-layered and multi-species forests in central Europe

  • Ram P. SharmaEmail author
  • Zdeněk Vacek
  • Stanislav Vacek
  • Miloš Kučera
Original Article
Part of the following topical collections:
  1. Biomechanics


Key message

The proposed height–diameter model applicable to many tree species in the multi-layered and mixed stands across Czech Republic shows a high accuracy in the height prediction. This model can be useful for estimating forest yield and biomass, and simulation of the vertical stand structures.


We developed a generalized nonlinear mixed-effects height–diameter (H–D) model applicable to Norway spruce (Picea abies (L.) Karst.), European beech (Fagus sylvatica L.) and other conifer and broadleaved tree species using the modelling method that includes dummy variables accounting for species-specific height differences and random component accounting for within- and between-sample plot height differences and randomness in the data. We used two large datasets: the first set (model fitting dataset) originated from permanent research sample plots and second set (model-testing dataset) originated from the Czech national forest inventory (NFI) sample plots. The former dataset comprises 224 sample plots with 29,390 trees and the latter dataset comprises 14,903 sample plots with 382,540 trees, each representing wide variabilities of tree size, ecological zone, growth condition, stand structure and development stage, and management regime across the country. Among the four versatile growth functions evaluated as base functions with diameter at breast height (DBH) included as a single predictor, the Chapman-Richards function showed the most attractive fit statistics. This function was then extended through the integration of other predictor variables, which better describe the stand density (stand basal area), stand development and site quality (dominant height), competition (ratio of DBH to quadratic mean DBH), that would act as modifiers of the original parameters of the function. The mixed-effects H–D model described a large part of the variations in the H–D relationships (\(R_{{{\text{adj}}}}^{2}\) = 0.9182; RMSE = 2.7786) without substantial trends in the residuals. Testing this model against model-testing dataset confirmed the model’s high accuracy. The model may be used for estimating forest yield and biomass, and therefore will serve as an important tool for decision making in forestry.


Chapman-Richards function National forest inventory Dummy variable modelling Stand density Stand structure Random effects Species-specific height difference 



This study was supported by Czech Ministry of Agriculture (No. QJ1520037), Faculty of Forestry and Wood Sciences, Czech University of Life Sciences Prague (IGA No. B2018, B0318, and Excellent Output 2018), and EXTEMIT-k project (No. CZ.02.1.01/0.0/0.0/15_003/0000433). We thank two anonymous reviewers for their constructive comments and insightful suggestions.

Author contributions

Ram Sharma: conceptualized and designed the study, performed data analysis and modelling, and wrote and revised the manuscript; Zdeněk Vacek: contributed to data descriptions and produced study area maps, and worked for revision, Stanislav Vacek: provided training dataset and contributed to the manuscript improvement. Miloš Kučera: provided validation dataset and contributed to the manuscript improvement.

Compliance with ethical standards

Conflict of interest

Authors have no conflict of interest.

Supplementary material

468_2018_1762_MOESM1_ESM.docx (956 kb)
Supplementary material 1 (DOCX 956 KB)
468_2018_1762_MOESM2_ESM.docx (19 kb)
Supplementary material 2 (DOCX 19 KB)
468_2018_1762_MOESM3_ESM.docx (13 kb)
Supplementary material 3 (DOCX 12 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Forestry and Wood SciencesCzech University of Life Sciences PraguePrague 6Czech Republic
  2. 2.Forest Management InstituteBrandýs nad Labem-Stará BoleslavCzech Republic

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