Height to crown base (HCB) of a tree is an important variable often included as a predictor in various forest models that serve as the fundamental tools for decision-making in forestry. We developed spatially explicit and spatially inexplicit mixed-effects HCB models using measurements from a total 19,404 trees of Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.) on the permanent sample plots that are located across the Czech Republic. Variables describing site quality, stand density or competition, and species mixing effects were included into the HCB model with use of dominant height (HDOM), basal area of trees larger in diameters than a subject tree (BAL- spatially inexplicit measure) or Hegyi's competition index (HCI-spatially explicit measure), and basal area proportion of a species of interest (BAPOR), respectively. The parameters describing sample plot-level random effects were included into the HCB model by applying the mixed-effects modelling approach. Among several functional forms evaluated, the logistic function was found most suited to our data. The HCB model for Norway spruce was tested against the data originated from different inventory designs, but model for European beech was tested using partitioned dataset (a part of the main dataset). The variance heteroscedasticity in the residuals was substantially reduced through inclusion of a power variance function into the HCB model. The results showed that spatially explicit model described significantly a larger part of the HCB variations [R2adj = 0.86 (spruce), 0.85 (beech)] than its spatially inexplicit counterpart [R2adj = 0.84 (spruce), 0.83 (beech)]. The HCB increased with increasing competitive interactions described by tree-centered competition measure: BAL or HCI, and species mixing effects described by BAPOR. A test of the mixed-effects HCB model with the random effects estimated using at least four trees per sample plot in the validation data confirmed that the model was precise enough for the prediction of HCB for a range of site quality, tree size, stand density, and stand structure. We therefore recommend measuring of HCB on four randomly selected trees of a species of interest on each sample plot for localizing the mixed-effects model and predicting HCB of the remaining trees on the plot. Growth simulations can be made from the data that lack the values for either crown ratio or HCB using the HCB models.

Faculty of Forestry and Wood Sciences Czech University of Life Sciences Prague Prague Suchdol Czech Republic

Krkonoše Mountains National Park Administration Department of Nature Conservation Dobrovského Vrchlabí Czech Republic

Zobrazit více v PubMed

Ritchie MW, Hann DW. Equations for predicting height to crown base for fourteen tree species in southwest Oregon. Oregon State University, Forestry Research Laboratory, Corvallis, OR: 1987.

Pearcy RW, Muraoka H, Valladares F. Crown architecture in sun and shade environments: assessing function and trade-offs with a three-dimensional simulation model. New Phytology. 2005; 166(3):791–800. PubMed

Buckley TN, Cescatti A, Farquhar GD. What does optimization theory actually predict about crown profiles of photosynthetic capacity when models incorporate greater realism? Plant, Cell & Environment. 2013; 36(8):1547–63. PubMed

Fu L, Zhang H, Sharma RP, Pang L, Wang G. A generalized nonlinear mixed-effects height to crown base model for Mongolian oak in northeast China. Forest Ecology and Management. 2017; 384:34–43.

Hasenauer H, Monserud RA. A crown ratio model for Austrian forests. Forest Ecology and Management. 1996; 84(1–3):49–60.

Assmann E. The principles of forest yield studies. Pergamon press, Oxford, 506 p. 1970.

Zarnoch SJ, Bechtold WA, Stolte KW. Using crown condition variables as indicators of forest health. Canadian Journal of Forest Research. 2004; 34(5):1057–70.

Kershaw JA Jr, Maguire DA, Hann DW. Longevity and duration of radial growth in Douglas-fir branches. Canadian Journal of Forest Research. 1990; 20(11):1690–5.

Kuprevicius A, Auty D, Achim A, Caspersen JP. Quantifying the influence of live crown ratio on the mechanical properties of clear wood. Forestry. 2014; 87 (3):449–58

Navratil S. Wind damage in thinned stands. In: Proceedings of a Commercial Thinning Workshop. October 17–18. Whitecourt, pp. 29–36. 1997.

Wykoff WR, Crookston NL, Stage AR. User’s guide to the stand prognosis model. Gen. Tech. Rep. INT-133. USDA, Forest Service, Intermountain Forest and Range Experiment Station, 112 p. 1982.

Sprinz PT, Burkhart HE. Relationships between tree crowns, stem, and stand characteristics in unthinned loblolly pine plantations. Canadian Journal of Forest Research. 1987; 17(6):534–8.

Wykoff WR. A basal area increment model for individual conifers in the Northern Rocky Mountain. Forest Science. 1990; 36(4):1077–104.

Stage AR. Prognosis model for stand development. USDA Forest Service, Intermountain Forest and Range Experiment Station, Ogden, Utah. Research Paper INT-137. 32 p. 1973.

Daniels RF, Burkhart HE. Simulation of individual tree growth and stand development in managed loblolly pine plantations. Division of Forestry and Wildlife Resources, Virginia Polytechnic and State University, Blacksburg: FWS-5-75, 69 p. 1975.

Ritchie MW, Hann DW. Equations for predicting basal area increment for Douglas-fir and grand fir Forest Research Laboratory, Oregon State University, Corvallis: Research Bulletin 51, 9 p. 1985.

Wensel LC, Koehler JR. A tree growth projection system for northern California coniferous forests Northern California Forest Yield Cooperative, Department of Forestry and Resource Management, University of California, Berkeley: Research Note 12, 30 p. 1985.

Walters DK, Hann DW. Taper equations for six conifer species in southwest Oregon Forest Research Laboratory, Oregon State University, Corvallis: Research Bulletin 56, 41 p. 1986.

McGaughey RJ. Visualizing forest stand dynamics using the stand visualization system. Proceedings of the 1997 ACSM/ASPRS Annual convention and exposition. April 7–10. Seattle, pp. 248–257. 1997.

Tews J, Brose U, Grimm V, Tielbörger K, Wichmann MC, Schwager M, et al. Animal species diversity driven by habitat heterogeneity/diversity: the importance of keystone structures. Journal of Biogeography. 2004; 31(1):79–92.

Maguire DM. Construction of regression models for predicting crown development in southwestern Oregon Douglas-fir. Ph.D. thesis. Oregon State University, 201 p. 1986.

Van Deusen PC, Biging GS. STAG, a STAnd Generator for mixed species stands Northern California Forest Yield Cooperative, Department of Forestry and Resource Management, University of California, Berkeley: Research Note 11, 25 p. 1985.

Temesgen H, LeMay V, Mitchell SJ. Tree crown ratio models for multi-species and multi-layered stands of southeastern British Columbia. The Forestry Chronicle. 2005; 81(1):133–41.

McRoberts RE, Hahn JT, Hefty GJ, Cleve JRV. Variation in forest inventory field measurements. Canadian Journal of Forest Research. 1994; 24(9):1766–70.

Rijal B, Weiskittel AR, Kershaw JA. Development of height to crown base models for thirteen tree species of the North American Acadian Region. The Forestry Chronicle. 2012; 88(1):60–73.

Hann DW, Hester AS, Olson CL. ORGANON User’s Manual. Department of Forest Resources, Oregon State University, 1997.

Russell MB, Weiskittel AR, Kershaw JA. Comparing strategies for modeling individual-tree height and height-to-crown base increment in mixed-species Acadian forests of northeastern North America. European Journal of Forest Research. 2014; 133(6):1121–35.

Marshall DD, Johnson GP, Hann DW. Crown profile equations for stand-grown western hemlock trees in northwestern Oregon. Canadian Journal of Forest Research. 2003; 33(11):2059–66.

Valentine HT, Amateis RL, Gove JH, Mäkelä A. Crown-rise and crown-length dynamics: application to loblolly pine. Forestry. 2013; 86(3):371–5.

Soares P, Tomé M. A tree crown ratio prediction equation for eucalypt plantations. Annals of Forest Science. 2001; 58(2):193–202.

Fu L, Zhang H, Lu J, Zang H, Lou M, Wang G. Multilevel nonlinear mixed-effect crown ratio models for individual trees of Mongolian Oak ( PubMed DOI PMC

Leites LP, Robinson AP, Crookston NL. Accuracy and equivalence testing of crown ratio models and assessment of their impact on diameter growth and basal area increment predictions of two variants of the Forest Vegetation Simulator. Canadian Journal of Forest Research. 2009; 39(3):655–65.

Zumrawi AA, Hann DW. Equations for predicting the height to crown base of six species in the Central Western Willamette Valley of Oregon. Oregon State University, Forest Research Laboratory, Corvallis, OR. Research Paper 52, 16 p. 1989.

Liu J, Burkhart HE, Amateis RL. Projecting crown Measures for loblolly pine trees using a generalized thinning response function. Forest Science 1995; 41(1):43–53.

Davies O, Pommerening A. The contribution of structural indices to the modelling of Sitka spruce (

Thorpe HC, Astrup R, Trowbridge A, Coates KD. Competition and tree crowns: A neighborhood analysis of three boreal tree species. Forest Ecology and Management. 2010; 259(8):1586–96.

Petritan AM, Von Lüpke B, Petritan IC. Effects of shade on growth and mortality of maple (

Long JN, Vacchiano G. A comprehensive framework of forest stand property–density relationships: perspectives for plant population ecology and forest management. Annals of Forest Science. 2014; 71(3):325–35.

Pretzsch H. Forest dynamics, growth and yield: from measurement to model. Springer Verlag, Berlin, Germany, 664 p. 2009.

Contreras MA, Affleck D, Chung W. Evaluating tree competition indices as predictors of basal area increment in western Montana forests. Forest Ecology and Management. 2011; 262(11):1939–49.

Sharma RP, Brunner A. Modeling individual tree height growth of Norway spruce and Scots pine from national forest inventory data in Norway. Scandinavian Journal of Forest Research. 2016:1–14.

Condés S, Del Rio M, Sterba H. Mixing effect on volume growth of

Sterba H, Rio M, Brunner A, Condes S. Effect of species proportion definition on the evaluation of growth in pure vs. mixed stands. Forest Systems 2014; 23(3):547–59.

Bayer D, Seifert S, Pretzsch H. Structural crown properties of Norway spruce (

Pretzsch H, Forrester DI, Rötzer T. Representation of species mixing in forest growth models. A review and perspective. Ecological Modelling. 2015; 313:276–92.

Sharma RP, Vacek Z, Vacek S. Individual tree crown width models for Norway spruce and European beech in Czech Republic. Forest Ecology and Management. 2016; 366:208–20.

Río M, Pretzsch H, Alberdi I, Bielak K, Bravo F, Brunner A, et al. Characterization of the structure, dynamics, and productivity of mixed-species stands: review and perspectives. European Journal of Forest Research. (Journal article). 2016; 135(1):23–49.

Pinheiro JC, Bates DM. Mixed-effects models in S and S-PLUS. New York: Springer; 2000.

Calama R, Montero G. Interregional nonlinear height-diameter model with random coefficients for stone pine in Spain. Canadian Journal of Forest Research 2004. January; 34(1):150–63.

Fu L, Sharma RP, Hao K, Tang S. A generalized interregional nonlinear mixed-effects crown width model for Prince Rupprecht larch in northern China. Forest Ecology and Management. 2017; 389:364–73.

Vonesh EF, Chinchilli VM. Linear and nonlinear models for the analysis of repeated measurements. New York: Marcel Dekker; 1997.

Lindstrom JM, Bates DM. Non-linear mixed effects models for repeated measures data. Biometrics. 1990; 46:673–87. PubMed

Calama R, Montero G. Multilevel linear mixed model for tree diameter increment in stone pine (

Meng SX, Huang SM, Yang YQ, Trincado G, VanderSchaaf CL. Evaluation of population-averaged and subject-specific approaches for modeling the dominant or codominant height of lodge pole pine trees. Canadian Journal of Forest Research. 2009; 39(6):1148–58.

Vacek Z, Vacek S, Podrázský V, Bílek L, Štefančík I, Moser WK, et al. Effect of tree layer and microsite on the variability of natural regeneration in autochthonous beech forests. Polish Journal of Ecology. 2015; 63(2):233–46.

Vacek Z, Vacek S, Bílek L, Remeš J, Štefančík I. Changes in horizontal structure of natural beech forests on an altitudinal gradient in the Sudetes. Dendrobiology. 2015; 73:33–45.

Sharma RP, Bílek L, Vacek Z, Vacek S. Modelling crown width-diameter relationship for Scots pine in the central Europe. Trees (in press). doi: 10.1007/s00468-017-1593-8 DOI

Sharma RP, Vacek Zk, Vacek S. Modeling individual tree height to diameter ratio for Norway spruce and European beech in Czech Republic. Trees. 2016; 30(6):1969–82.

FMI. Inventarizace lesů, Metodika venkovního sběru dat (Forest inventory, field data collection methodology). Brandýs nad Labem, 136 p. 2003.

Sharma RP, Vacek Z, Vacek S, Jansa V. Modelling individual tree diameter growth for Norway spruce in Czech Republic using generalized algebraic difference approach. Journal of Forest Science. 2017; 63(5):227–238.

Taylor JE, Ellis MV, Rayner L, Ross KA. Variability in allometric relationships for temperate woodland Eucalyptus trees. Forest Ecology and Management. 2016; 360:122–32.

Bollandsås OM, Næsset E. Weibull models for single-tree increment of Norway spruce, Scots pine, birch and other broadleaves in Norway. Scandinavian Journal of Forest Research. 2009; 24(1):54–66.

Gill SJ, Biging GS, Murphy EC. Modeling conifer tree crown radius and estimating canopy cover. Forest Ecology and Management. 2000; 126(3):405–16.

Ritson P, Sochacki S. Measurement and prediction of biomass and carbon content of

Fu L, Sun H, Sharma RP, Lei Y, Zhang H, Tang S. Nonlinear mixed-effects crown width models for individual trees of Chinese fir (

Sharma RP, Brunner A, Eid T, Øyen B-H. Modelling dominant height growth from national forest inventory individual tree data with short time series and large age errors. Forest Ecology and Management. 2011; 262(12):2162–75.

Zhao D, Kane M, Borders BE. Crown ratio and relative spacing relationships for loblolly pine plantations. Open Journal Forestry. 2012; 2(3):101–15.

Biging GS, Dobbertin M. Comparison of distance-dependent competition measures for height and basal area growth of individual conifer trees. Forest Science. 1992; 38(3):695–720.

Pretzsch H, Biber P, Dursky J. The single tree-based stand simulator SILVA: construction, application and evaluation. Forest Ecology and Management. 2002; 162(1):3–21.

Hegyi F. A simulation model for managing jack-pine stands In: Fries J, editor. Growth models for tree and stand simulation: Royal College of Forestry, Stockholm, Sweden: Research Note 30; 1974. p. 74–90.

Martin GL, Ek AR, Monserud RA. Control of plot edge bias in forest stand growth simulation models. Canadian Journal of Forest Research. 1977; 7(1):100–5.

Goreaud F, Pélissier R. On explicit formulae of edge effect correction for Ripley's K-function. Journal of Vegetation Science. 1999; 10(3):433–8.

Uzoh FCC, Oliver WW. Individual tree diameter increment model for managed even-aged stands of ponderosa pine throughout the western United States using a multilevel linear mixed effects model. Forest Ecology and Management. 2008; 256(3):438–45.

Montgomery DC, Peck EA, Vining GG. Introduction to linear regression analysis. 3

Sharma RP, Brunner A, Eid T. Site index prediction from site and climate variables for Norway spruce and Scots pine in Norway. Scandinavian Journal of Forest Research. 2012; 27(7):619–36.

SAS Institute Inc. SAS/ETS1 9.1.3 User’s Guide. SAS Institute Inc., Cary, NC: 2012.

Littell RC, Milliken GA, Stroup WW, Wolfinger RD, Schabenberger O. SAS for mixed models, 2

Hirsch RP. Validation samples. Biometrics 1991; 47:1193–4. PubMed

Kozak A, Kozak R. Does cross validation provide additional information in the evaluation of regression models? Canadian Journal of Forest Research. 2003; 33(6):976–87.

Yang YQ, Monserud RA, Huang SM. An evaluation of diagnostic tests and their roles in validating forest biometric models. Canadian Journal of Forest Research. 2004; 34(3):619–29.

Sharma M, Parton J. Height-diameter equations for boreal tree species in Ontario using a mixed-effects modeling approach. Forest Ecology and Management. 2007; 249(3):187–98.

Yang YQ, Huang SM, Trincado G, Meng SX. Nonlinear mixed-effects modeling of variable-exponent taper equations for lodgepole pine in Alberta, Canada. European Journal of Forest Research. 2009; 128(4):415–29.

Huang S, Meng SX, Yang Y. Using nonlinear mixed model technique to determine the optimal tree height prediction model for black spruce. Modern Applied Science. 2009; 3(4):3–18.

Sharma RP, Breidenbach J. Modeling height-diameter relationships for Norway spruce, Scots pine, and downy birch using Norwegian national forest inventory data. Forest Science and Technology. 2015; 11(1):44–53.

Podlaski R. Highly skewed and heavy-tailed tree diameter distributions: approximation using the gamma shape mixture model. Canadian Journal of Forest Research. 2016; 46(11):1275–83.

Podlaski R. Forest modelling: the gamma shape mixture model and simulation of tree diameter distributions. Annals of Forest Science. 2017/ April/ 03; 74(2):29.

Vanclay JK. Modelling forest growth and yield Applications to mixed tropical forests. CAB International, Oxon, U.K., p. 312; 1994.

Hasenauer H. Concepts within tree growth modeling In: Hasenauer H. (Ed.) Sustainable forest management: Growth models for Europe, Springer Verlag, Berlin Heidelberg, 398 p. 2006.

Canham CD, LePage PT, Coates KD. A neighborhood analysis of canopy tree competition: effects of shading versus crowding. Canadian Journal of Forest Research. 2004; 34(4):778–87.

Short III EA, Burkhart H. Prediction crown-height increment for thinned and unthinned loblolly pine plantations Forest Science. 1992; 38 594–610.

Hynynen J. Predicting the growth response to thinning for Scots pine stands using individual-tree growth models. Silva Fennica. 1995; 29 225–46.

Monserud RA. Height growth and site index curves for inland Douglas-fir based on stem analysis data and forest habitat type. Forest Science. 1984; 30(4):943–65.

Russell MB, Weiskittel AR. Maximum and Largest Crown Width Equations for 15 Tree Species in Maine. Northern Journal of Applied Forestry. 2011; 28(2):84–91.

Toney C, Reeves C. Equations to convert compacted crown ratio to uncompacted crown ratio for trees in the interior West. Western Journal of Applied Forestry. 2009; 24(2):76–82.

Power H, LeMay V, Berninger F, Sattler D, Kneeshaw D. Differences in crown characteristics between black (

Sorrensen-Cothern KA, Ford ED, Sprugel DG. A model of competition incorporating plasticity through modular foliage and crown development. Ecological Monograph. 1993; 63(3):277–304.

Kantola A, Mäkinen H, Mäkelä A. Stem form and branchiness of Norway spruce as a sawn timber-Predicted by a process based model. Forest Ecology and Management. 2007; 241(1–3):209–22.

Putz FE, Parker G, Archibald M. Mechanical abrasion and inter-crown spacing. American Middle Nature. 1984; 112(1):24–8.

Bragg DC. A local basal area adjustment for crown width prediction. Northern Journal of Applied Forestry. 2001; 18(1):22–8.

Corral-Rivas JJ, Gonzalez JGA, Aguirre O, Hernandez F. The effect of competition on individual tree basal area growth in mature stands of

Mailly D, Turbis S, Pothier D. Predicting basal area increment in a spatially explicit, individual tree model: a test of competition measures with black spruce. Canadian Journal of Forest Research. 2003; 33(3):435–43.

Porte A, Bartelink HH. Modelling mixed forest growth: a review of models for forest management. Ecological Modelling. 2002; 150(1–2):141–88.

Bachmann M. About the effects of competition on individual tree growth in mountain forests. Allgemeine Forst und Jagdzeitung. 1997; 168(6–7):127–30.

Richards M, McDonald AJS, Aitkenhead MJ. Optimization of competition indices using simulated annealing and artificial neural networks. Ecological Modelling. 2008; 214(2–4):375–84.

Hyyppa J, Holopainen M, Olsson H. Laser scanning in forests. Remote Sensing. 2012; 4(10):2919–22.

Pretzsch H. Analysis and modeling of spatial stand structures. Methodological considerations based on mixed beech-larch stands in Lower Saxony. Forest Ecology and Management. 1997; 97(3):237–53.

Jucker T, Bouriaud O, Coomes DA. Crown plasticity enables trees to optimize canopy packing in mixed-species forests. Functional Ecology. 2015; 29(8):1078–86.

Olivier M-D, Robert S, Fournier RA. Response of sugar maple (

Temesgen H, Monleon VJ, Hann DW. Analysis and comparison of nonlinear tree height prediction strategies for Douglas-fir forests. Canadian Journal of Forest Research. 2008; 38(3):553–65.

Sharma RP, Vacek Z, Vacek S. Nonlinear mixed effect height-diameter model for mixed species forests in the central part of the Czech Republic. Journal of Forest Science. 2016; 62(10):470–84.

Crecente-Campo F, Tomé M, Soares P, Dieguez-Aranda U. A generalized nonlinear mixed-effects height-diameter model for