Forestry & Forest Sciences · Mensuration, Inventory & Geospatial Analysis · also Forest Growth and Yield, Yield Tables, Forest Modelling
Predicting how stands grow and what they will yield: yield tables, stand density, diameter-distribution and individual-tree models, and their use in planning.
In the exams: ICAR AICE-JRF 59 Unit 4 (growth and increment, yield) and Unit 9 (growing stock, site quality); IFoS Forestry Paper II Section A; SAF Certified Forester
Growth and yield modelling is the branch of forest science that predicts how trees and stands grow and what they will yield. Growth is the increase in size over a period; yield is the size reached at a given time, or the amount that can be harvested. A growth and yield model takes what is known about a stand today, usually its age, site quality, density and the sizes of its trees, and projects it forward under a stated management regime. Its outputs answer the questions every forest manager faces: how much timber will be available, when the stand should be thinned or harvested, what a change in spacing or rotation will do, and how much carbon the forest will hold.
Models range from simple to complex. The oldest are yield tables, printed tables of volume and other stand values by age and site quality, built for fully stocked or average stands. Next come stand density measures and the self-thinning relationships that describe the limits of crowding, and density management diagrams that turn those limits into thinning schedules. Whole-stand models predict stand totals from equations; diameter-distribution models add the number of trees in each size class; individual-tree models grow each tree separately, with or without its position; and process-based models simulate photosynthesis, respiration and allocation from climate and soil. All share the same discipline of fitting to data, validating against independent data and projecting with care.
This subject builds on two others in this track. Forest mensuration supplies the measurements and the basic concepts of increment, mean and current annual increment, and site index; inventory and sampling design are in forest-inventory-sampling. This manual does not repeat those but uses them. Forest management and yield regulation, which uses growth and yield predictions to set the allowable cut, is the next subject in the track.
In examinations, growth and yield sits in Unit 4 (growth and increment, yield) and Unit 9 (growing stock, site quality) of the ICAR AICE-JRF forestry syllabus, in Section A of Paper II of the Indian Forest Service forestry optional, and in the forest management and inventory content of professional credentials such as the SAF Certified Forester. Typical questions ask for definitions of the kinds of yield table, the properties of growth functions, the calculation of stand density index, relative spacing or crown competition factor, the meaning of the self-thinning line, the difference between distance-dependent and distance-independent models, and the statistics used to validate a model. The numerical archetypes are computed in this manual and in the question bank.
When the cumulative size of a tree or an even-aged stand is plotted against age, the curve is sigmoid: slow at first, then rising steeply, then flattening towards an upper limit. The slope of this yield curve at any age is the current growth rate (current annual increment); the slope of a line from the origin to the curve is the mean annual increment. Growth rate rises to a maximum at the point of inflection of the yield curve and then declines. The upper limit, or asymptote, is the size the tree or stand would approach if it lived indefinitely; it depends on species and site.
A growth function is a mathematical equation that reproduces this shape. A good growth function for forestry should pass through zero at age zero (or at a defined starting point), rise to an asymptote, have an inflection point that can move to suit different species, and have parameters with biological meaning. Many have been proposed; a handful dominate forestry practice.
The Chapman-Richards function, Y = A(1 − e^(−kt))^c, is the most widely used in forestry. A is the asymptote, k a rate parameter and c a shape parameter that sets the position of the inflection point. It generalises the von Bertalanffy function (c = 3) and includes the monomolecular function (c = 1), which has no inflection point and is used for growth that slows from the start. With A = 30 m, k = 0.05 and c = 1.5, the predicted height at age 20 is 30 × (1 − e^(−1))^1.5 = 15.1 m.
The logistic function, Y = A / (1 + b e^(−kt)), is symmetric about its inflection point, which lies at half the asymptote. The Gompertz function, Y = A exp(−b e^(−kt)), has its inflection at A/e, about 37% of the asymptote, so growth peaks earlier relative to final size. The Schumacher function, ln Y = a − b/t, is linear in the reciprocal of age and was much used for yield tables because it can be fitted by ordinary linear regression after transformation: with a = 6.0 and b = 20, yield at age 40 is e^(6.0 − 0.5) = e^5.5 = 245. The Korf and Weibull-type functions are also used.
Polynomials can fit data well within their range but behave badly outside it, curving downwards or rising without limit, so they are poor choices for projection.
The value of a growth function in practice lies in parameters that mean something. In the Chapman-Richards height function, the asymptote A reflects site: better sites have higher A. The rate parameter k reflects how quickly the curve approaches A: fast-growing species and favourable climates have larger k. The shape parameter c moves the inflection point: with larger c the curve stays flat longer before rising, and the growth rate peaks later. For c greater than 1 the inflection lies at A(1 − 1/c)^c; for c = 3 that is about 30% of the asymptote. When site-index curves are built from this function, either A or k (or both) is made to depend on site index, giving anamorphic curves if only A varies and polymorphic curves if the shape changes as well. Parameters are estimated by non-linear least squares, which needs starting values; poor starting values can make fitting fail or converge to a nonsensical solution, so biologically sensible starting values (for example an asymptote near the largest observed height) are chosen.
Because growth is the rate of change of yield, a yield equation and a growth equation should be mathematically consistent: integrating the growth equation over a period should give the change in the yield equation. Models with this property are called compatible. Early yield tables and separately fitted growth equations were often incompatible, giving different answers for the same stand depending on which was used. Clutter, in 1963, showed how to derive growth and yield equations that are compatible by construction, and the principle is now standard.
Growth functions are used for heights (site-index curves), diameters, basal areas, volumes and biomass of trees and stands. Fitted to height of dominant trees, the Chapman-Richards function generates site-index curves; with the asymptote or rate parameter made a function of site index, the same equation produces a family of curves for all sites (see the site-index chapter of forest-mensuration).
A yield table gives, for an even-aged stand of a given species, the expected values of stand variables at successive ages, usually in five- or ten-year steps, for each site quality class or site index. The variables typically include stems per hectare, mean and top height, quadratic mean diameter, basal area, volume (total and merchantable), mean annual increment and current or periodic annual increment, and sometimes the yield of thinnings. Yield tables were the first growth and yield models, developed in Germany in the eighteenth and nineteenth centuries, and they remain in use for many species, including the yield tables of the Forest Research Institute in India and the Forestry Commission's tables in Britain.
A normal yield table describes fully stocked stands, called normal stands, in which the trees fully occupy the site. It is built from temporary sample plots chosen in stands judged to be fully stocked, across a range of ages and sites. Because the stands represent an ideal of full stocking, the table gives the yield a well-managed stand should achieve.
To apply a normal yield table to a real stand, its stocking relative to normal must be estimated. Stocking (or normality) is commonly expressed as the ratio of the stand's actual basal area (or volume) to the normal basal area at the same age and site: a stand of 21 m²/ha where the table shows 30 m²/ha is 70% stocked. The table's volume is then multiplied by this fraction. The weakness is the assumption that understocked stands remain understocked by the same proportion, whereas in reality they tend to approach normal stocking over time, and judging which stands are normal is subjective.
An empirical yield table describes the average stands actually found, not ideal ones. It is built from sample plots in a representative range of stands, whatever their stocking, so it predicts the yield of a typical stand in the region. Empirical tables avoid the subjectivity of choosing normal stands but apply only to stands of average density; they cannot predict the effect of changing density by thinning or spacing.
A variable-density yield table includes stand density as an explicit variable, alongside age and site. MacKinney and Chaiken's work on loblolly pine in 1939 is usually cited as the first. Density is entered as basal area, stems per hectare or a stand density index, and yields are given for a range of densities. This allows the effect of thinning and spacing to be predicted directly. Variable-density tables are usually expressed as equations rather than printed tables: they are the first whole-stand growth and yield models in the modern sense (see the whole-stand models chapter).
Managed-stand yield tables, such as the British management tables, describe stands under a stated thinning regime, with the volume removed in thinnings listed separately from the main crop. They are useful for planning but apply only to the regime they describe.
The classical construction of a normal yield table follows a sequence that is still instructive. Temporary sample plots are laid out in stands judged fully stocked, covering the range of ages and sites; on each, stems per hectare, diameters, heights of dominant trees and volumes are measured. Site classes are defined from the height of dominant trees against age, usually by a guide curve and proportional curves, so that each plot can be assigned a site class. Within each site class, stand variables are plotted or regressed against age and smoothed curves drawn: top height, mean diameter, stems per hectare, basal area and volume. Mean annual increment is computed as volume divided by age, and periodic annual increment from the differences between successive tabulated volumes. The tables are then checked for internal consistency, for example that basal area equals stems times the basal area of the mean tree, and that increment curves have the expected shape, with current increment peaking before mean increment.
Modern practice fits equations to permanent-plot data instead of drawing curves by hand, and prints tables from the equations if printed tables are still wanted. The table and the equation then carry the same information, and the equation can be used directly in software.
Yield tables are used to estimate current and future volume, to set rotation ages at the culmination of mean annual increment, to construct normal growing stocks for yield regulation, and as benchmarks for actual growth. They should be applied only to the species, region and silvicultural regime for which they were built. Old tables may not reflect current genetic material, silviculture or climate; plantations of improved stock often grow faster than tables built from older stands. A table built from temporary plots of different ages assumes that stands of different ages on the same site class follow one growth path, which may not hold if site quality is not evenly represented across ages.
Stand density is the degree of crowding of trees on a site. It governs competition, individual tree growth, mortality, crown development, wood quality and the total volume produced. Silviculturists control it through planting spacing and thinning, and growth models need it as an input. Simple measures are stems per hectare (N) and basal area per hectare (G). Each has a weakness: N ignores tree size, and G treats many small trees and a few large ones as equivalent. Relative measures compare a stand with a reference, such as the maximum density possible for its mean tree size.
Reineke observed in 1933 that fully stocked, even-aged stands of many species follow a straight line when the logarithm of stems per unit area is plotted against the logarithm of quadratic mean diameter, with a slope of about −1.605. His stand density index (SDI) expresses any stand's density as the equivalent number of trees per unit area at a reference diameter of 10 inches (25.4 cm). In metric units, SDI = N × (Dq / 25.4)^1.605, with N per hectare and Dq in centimetres. A stand of 900 stems/ha with a Dq of 20 cm has SDI = 900 × (20 / 25.4)^1.605 = 613 trees/ha at the reference size. In US units, SDI = N × (Dq / 10)^1.605 with N per acre and Dq in inches.
Each species has a maximum SDI, observed in the densest stands. Relative density is SDI divided by maximum SDI. Long (1985) proposed widely cited thresholds: about 25% of maximum for the onset of competition (crown closure), about 35% for the lower limit of full site occupancy, and about 60% for the onset of self-thinning, where competition-induced mortality begins. Silviculturists keep stands between roughly 35% and 60% to use the site fully while avoiding mortality. SDI is largely independent of age and site, which makes it useful across stands.
Relative spacing is the average distance between trees divided by the dominant (top) height. Expressed as a percentage, it is often called the Hart-Becking spacing index. For square spacing, average distance is √(10 000 / N) metres. A stand of 1111 stems/ha (3 m square) with top height 20 m has a relative spacing of 3 / 20 = 0.15, or 15%. As a stand grows taller without thinning, relative spacing falls; thinning raises it. Many thinning regimes, particularly in Europe and for plantations, are expressed as a target relative spacing. Other spacing formulas assume triangular spacing, giving a slightly different average distance for the same N, so the convention must be stated.
Crown competition factor (CCF), introduced by Krajicek, Brinkman and Gingrich in 1961, compares the crown area the trees would have if open-grown with the area available. For each tree, the maximum crown area (MCA) of an open-grown tree of the same diameter is calculated from a species-specific equation; CCF is 100 times the sum of MCAs divided by the plot area. A CCF of 100 means that the open-grown crowns of the trees would just cover the ground; values above 100 mean crowns must be restricted. If 1000 stems/ha each have an open-grown crown area of 12 m², CCF = 100 × 12 000 / 10 000 = 120. CCF is useful in mixed and uneven-aged stands and as a predictor in individual-tree models.
In crowded even-aged stands, trees die as the survivors grow, and the combinations of mean size and density that stands can reach are bounded. Reineke's line is one expression: in fully stocked stands, log N falls linearly with log Dq, with slope about −1.6. If Dq doubles along the line, N falls by a factor of 2^1.605, about 3. In plant ecology, Yoda and colleagues described in 1963 the −3/2 power rule: in self-thinning populations, mean plant mass is proportional to density raised to the power −3/2. The two are consistent if mass scales with diameter to a power of about 2.4.
The exact slope varies somewhat between species and studies, and the line represents a maximum boundary that stands approach but do not cross, not a path that all stands follow. Stands at low density grow along a line of nearly constant N until they approach the boundary, then bend and move along it as self-thinning begins.
A density management diagram (DMD) is a graphical model that brings the self-thinning boundary and stand growth into one chart for a species. Developed in Japan by Ando and colleagues in the 1960s and widely used in North America after Drew and Flewelling's work in 1979, it typically plots the logarithm of stems per hectare against the logarithm of mean tree volume or quadratic mean diameter. The chart carries the maximum-density (self-thinning) line, lines of equal relative density (for example the 25%, 35% and 60% thresholds), and isolines of dominant height and Dq.
A manager uses a DMD to plan a regime. Starting from the planted density, the stand's trajectory is traced as height increases; when it reaches an upper target relative density, a thinning is drawn as a drop to a lower target density at constant height; growth then resumes. The chart shows the Dq and volume at each stage and the final harvest. DMDs are simple, visual and well suited to plantations, but they describe average stands and do not show variation among trees.
Self-thinning relationships define the maximum carrying capacity of a site for a given tree size, set upper limits in growth models, give reference values for relative density, and guide spacing and thinning. They are most reliable in even-aged, single-species stands. Mixed stands, uneven-aged stands and stands under strong site limitations may follow different boundaries. Climate change and atmospheric CO2 may also shift maximum densities, which is a current research topic.
A whole-stand model predicts stand-level variables, such as volume, basal area and stems per hectare, directly from stand-level inputs: age, site index and a density measure. It does not track individual trees or size classes, although some predict the mean diameter. Whole-stand models are the equation form of yield tables, extended to include density; they are compact, need little data to run, and are well suited to plantations and even-aged stands managed in large units.
Whole-stand models are classified by how density enters. Models with fixed density (normal or empirical yield tables) have no density variable. Variable-density models include it: either explicitly, as an input such as basal area or stems per hectare at the time of prediction, or implicitly, as an initial density that the model projects forward. A typical explicit model has the Schumacher form ln V = b0 + b1/A + b2 S + b3 ln G, predicting volume V from age A, site index S and basal area G. Because current basal area must itself be known, a companion equation projects basal area from one age to another, for example ln G2 = (A1/A2) ln G1 + b (1 − A1/A2) + c (1 − A1/A2) S.
Such projection equations are path invariant if projecting directly from age A1 to A3 gives the same result as projecting from A1 to A2 and then from A2 to A3. Path invariance and compatibility between growth and yield equations are desirable properties of any projection system.
Whole-stand models are robust and easy to fit to permanent-plot data, and they are accurate for total stand volume. They are less useful when the distribution of tree sizes matters, for example when products depend on log size, when thinnings remove particular size classes, or in uneven-aged and mixed stands. Diameter-distribution and individual-tree models address these needs. Many national plantation-planning systems still rely on whole-stand models because they are dependable and cheap to apply.
Product value depends on tree size: sawlogs, veneer logs and poles need trees above certain diameters, while pulpwood can come from small trees. A stand's total volume does not say how it is split among sizes. Diameter-distribution models predict the number of trees in each diameter class, from which heights, volumes and products are calculated class by class using height-diameter curves and volume or taper equations. They sit between whole-stand models and individual-tree models in detail.
A probability distribution is fitted to the diameters in each stand, and its parameters are related to stand variables. The Weibull distribution is the most widely used because it is flexible and its cumulative distribution has a closed form. The three-parameter Weibull has location a (the smallest possible diameter), scale b and shape c. The proportion of trees with diameter less than x is F(x) = 1 − exp(−((x − a)/b)^c). With a = 5 cm, b = 15 cm and c = 2, the proportion below 20 cm is 1 − e^(−1) = 0.632, or 63.2%.
The shape parameter controls form: c < 1 gives a reverse-J shape typical of uneven-aged stands; c = 1 is the exponential distribution; values around 3.6 give a nearly symmetric, bell-like shape typical of even-aged stands; higher values give left-skewed distributions. The Johnson SB, beta, gamma and lognormal distributions are also used.
Two approaches link the distribution to stand variables. In the parameter prediction method, the Weibull parameters fitted to each sample plot are regressed directly on stand variables such as age, site index and stems per hectare. In the parameter recovery method, stand-level attributes predicted by a whole-stand model, such as quadratic mean diameter, arithmetic mean diameter or percentiles of the distribution, are used to solve for the parameters, so that the distribution is consistent with the whole-stand prediction. Parameter recovery ensures that, for example, basal area computed from the distribution equals basal area predicted by the stand-level equation. Non-parametric approaches, such as percentile-based methods, are alternatives.
Diameter-distribution models are widely used for even-aged plantations, particularly where product breakdown and thinning by size class must be predicted. They are less suited to stands whose distributions are irregular or multimodal, such as mixed-species or recently thinned stands, unless more flexible distributions are used.
An individual-tree model grows each tree in a tree list separately, predicting its diameter growth, height growth and probability of death over a period, typically five or ten years. Stand values are obtained by summing over trees. Individual-tree models can represent any stand structure, mixed species, uneven ages and complex thinnings, and they are the core of many modern forest simulators, such as the Forest Vegetation Simulator used by the US Forest Service. They need tree lists as input, usually from inventory plots.
In distance-independent models, a tree's growth depends on its own size and on stand-level or relative measures of competition, without the positions of trees. Typical predictors are the tree's diameter and crown ratio, site index, stand basal area, and the basal area of trees larger than the subject tree (BAL), which measures how much larger competition overtops it. A tree's relative size in the stand, such as its diameter divided by Dq, is another common index. These models need only standard inventory data and are widely used in large-scale forest planning.
Distance-dependent (spatially explicit) models use the positions of trees, from stem maps, to compute competition from neighbours. Competition indices include Hegyi's index, the sum over competitors of the ratio of the competitor's diameter to the subject tree's diameter divided by the distance between them: CI = Σ (Dj / Di) / distij. For a subject tree of 20 cm with competitors of 25 cm at 4 m and 15 cm at 3 m, CI = (25/20)/4 + (15/20)/3 = 0.56. Other indices are based on overlap of influence zones around trees, on the area potentially available to each tree (Voronoi or Thiessen polygons), or on the angles to the tops of neighbours.
Distance-dependent models are in principle more realistic, especially in irregular stands, and are valuable for research on competition and spatial silviculture. In practice they often predict stand totals only slightly better than distance-independent models, while needing stem maps that are expensive to collect. Terrestrial and airborne laser scanning, which can map tree positions over large areas, may change this balance.
An individual-tree simulator is a set of linked sub-models. Besides diameter growth and mortality, it usually contains a height-diameter equation to fill in heights not measured in the inventory, a height-growth equation (often linked to site-index curves for dominant trees), a crown-ratio or crown-recession equation, volume and taper equations to convert trees into products, and rules for regeneration. Each sub-model is fitted separately, so errors in one propagate through the others; a biased height-growth equation, for example, distorts volume predictions even if diameter growth is right. Sub-models also need to be consistent with stand-level limits, so many simulators impose a maximum stand density (often from the self-thinning line) that triggers extra mortality when it is exceeded.
Because individual-tree simulators carry each tree, they can mimic almost any treatment: removing trees by species, size or crown class, leaving retention trees, or simulating selection cutting. Their outputs feed product estimates, habitat indices such as numbers of large trees and snags, and carbon accounts.
Individual-tree models must also predict mortality and, in uneven-aged stands, ingrowth (new trees crossing the minimum diameter). Mortality is usually modelled as a probability, with logistic regression on tree size, growth rate, crown ratio and competition. Because mortality is a rare event in any short period, large datasets from remeasured permanent plots are needed to estimate it well. Ingrowth models predict the number and species of new trees from stand density and composition. Errors in these components accumulate over long projections and are often the weakest part of individual-tree simulators.
The models in the previous chapters are empirical: they describe statistical relationships observed in data and assume that the future resembles the conditions of the past under which the data were collected. Process-based (mechanistic) models instead simulate the physiological processes that produce growth: light interception, photosynthesis, respiration, water use, nutrient uptake and the allocation of carbon to leaves, stems and roots, driven by weather, soil and atmospheric CO2. Because they represent causes, they can in principle predict growth under conditions not seen in the data, such as a changed climate or a new fertiliser regime.
The 3-PG model (Physiological Principles Predicting Growth), published by Landsberg and Waring in 1997, is the best-known process-based model used in forest management. It calculates gross primary production from absorbed light using a light-use efficiency, reduced by modifiers for temperature, frost, vapour pressure deficit, soil water, nutrition and stand age, and allocates the result to foliage, stems and roots with empirical rules. It runs on monthly weather and needs a modest set of parameters, which has made it popular for plantations of eucalypts and pines worldwide.
Gap models simulate the establishment, growth and death of individual trees on small patches, typically the size of a canopy gap, under climate and light constraints; JABOWA (1972) and FORET are early examples. They are used mainly to study long-term succession and the effects of climate change on species composition rather than to predict timber yield. Ecosystem models, which include soil carbon and nutrient cycles, are used in carbon accounting and climate research.
Process-based models are data-hungry, need parameters that are hard to measure, and are often less accurate than empirical models for predicting yield under current conditions. Hybrid models combine the two: an empirical growth model is modified by physiologically based factors, for example a site-productivity term predicted from climate and soil by a process model, or growth multipliers that respond to temperature and water stress. Hybrids keep the accuracy of empirical models under current conditions while gaining some sensitivity to environmental change. They are an active area of development as managers seek models that remain valid under climate change.
No one type is best. Choice depends on the question, the data and the scale. For predicting next decade's volume in a plantation estate, a whole-stand or diameter-distribution model is usually best; for mixed, uneven-aged forests with complex treatments, an individual-tree model; for climate-change scenarios or new sites without growth data, a process-based or hybrid model. The model's validation history in conditions like those where it will be used matters more than its sophistication.
Growth and yield models are fitted to data from permanent sample plots (remeasured at intervals), temporary plots (measured once, with growth reconstructed from increment cores or stem analysis), and designed experiments such as spacing and thinning trials. Permanent plots give the most reliable growth data because they measure the same trees repeatedly, but they take decades to accumulate. Temporary plots of different ages substitute space for time; they are quick but can confound site with age. Designed experiments give the widest range of densities and treatments, which observational plots rarely cover.
Models are fitted by regression: linear least squares for linear or linearised models, non-linear least squares for growth functions such as the Chapman-Richards. Repeated measurements of the same plots are correlated, which violates the independence assumption of ordinary regression; mixed-effects models, with random effects for plots or trees, handle this and allow the model to be calibrated to a new stand with a few local measurements. Heteroscedasticity, the increase in variance with size, is handled by weighting or transformation. Systems of equations that share variables, such as basal area and volume, may be fitted simultaneously so that their errors are consistent.
A model must be tested before it is trusted. Validation compares predictions with observations, ideally from independent data not used in fitting; when independent data are scarce, cross-validation (repeatedly fitting on part of the data and testing on the rest) is used. Common statistics are the bias, the mean of the residuals (observed minus predicted), which shows systematic over- or under-prediction; the root mean square error (RMSE), the square root of the mean squared residual, which combines bias and random error; and the model efficiency, analogous to R². For residuals of 2, −1, 3, −2 and 0, the bias is 0.4 and the RMSE is √3.6 = 1.90.
Residuals should be examined against age, site, density and tree size, because a model may be unbiased overall yet biased in some part of its range. Biological reasonableness matters too: predictions should not show negative growth, impossible densities or volumes that exceed any observed maximum.
Projection runs a model forward from a known starting point. Good projection systems are compatible (growth and yield equations agree) and path invariant (one long step equals several short ones). Errors compound over long projections, so uncertainty should be reported and projections periodically updated with new inventory data. Extrapolation, using a model outside the range of ages, sites, densities or treatments in its data, is the most common cause of serious error. Climate change adds a new form of extrapolation: empirical models fitted to past conditions may be biased in future ones, which is one reason for interest in hybrid models.
Growth and yield models predict individual stands; forest management needs predictions for the whole forest over decades. The link is the yield stream: for each stand, or each stratum of similar stands, the model is run under a set of candidate management regimes (for example different thinning schedules and rotation ages) to produce the volume, value and other outputs in each period. These yield streams are the coefficients of a harvest-scheduling model, which chooses a regime for each stand to meet the forest's objectives and constraints.
Most harvest-scheduling models are formulated as linear programmes. In the formulation known as Model I, each decision variable is the area of a stand or stratum assigned to one complete regime over the planning horizon, with its yields in every period taken from the growth model. In Model II, decision variables represent the area moving from one age class and period to the next, so regenerated stands form new units; it is more compact for long horizons. The objective is commonly to maximise net present value or volume, subject to constraints such as non-declining or even flow of harvest volume, a minimum ending inventory, area limits and, increasingly, spatial and environmental constraints. Forest management and yield regulation, the next subject in this track, covers these methods and the allowable cut in detail.
Errors in growth predictions pass directly into harvest schedules. If a model overestimates growth, the scheduled harvest will exceed what the forest can sustain, and the growing stock will decline. The allowable cut effect, the immediate increase in today's permissible harvest when expected future growth is raised (for example by planting improved stock), depends entirely on growth projections being right. For this reason, harvest schedules are revised regularly as new inventory data replace projections, and sensitivity analysis tests how the plan changes if growth is lower or higher than predicted.
Beyond harvest scheduling, growth models are used to compare silvicultural regimes, value forests and timberland for sale or taxation, estimate carbon stocks and sequestration for climate reporting and carbon projects, assess the effects of pests, fire or climate change, and train students and managers through simulation. Carbon projects in particular depend on credible baselines and projections, and their methodologies usually require models that have been validated for the region.
Learn the shape of growth curves and the meaning of each parameter in the Chapman-Richards function; the three kinds of yield table and how to correct a normal table for stocking; the formulas for SDI, relative spacing and CCF; what the self-thinning line means and how a density management diagram is read; and the differences between whole-stand, diameter-distribution, individual-tree and process-based models. Practise the numericals with a calculator, and for descriptive answers explain the trade-offs: why a variable-density model is better than a normal yield table for thinning decisions, or why distance-dependent models are not always worth their cost.
A forester five years in uses growth models routinely: projecting inventory data to the next working-plan period, choosing spacing and thinning regimes with a density management diagram or simulator, estimating when stands will reach merchantable size, preparing carbon estimates for project reports, and checking projections against remeasured plots. Practical skills include choosing a model validated for the species and region, recognising when a stand lies outside a model's data range, and maintaining permanent sample plots so that local growth can be checked.
A senior forester relies on growth projections for allowable cut and long-term planning across an estate or a forest division. Decisions include commissioning new or recalibrated models when genetics, silviculture or climate change, insisting on validation and sensitivity analysis before major harvest commitments, and maintaining the permanent-plot networks that make models credible. Over-optimistic growth assumptions have caused over-cutting in many forests; senior staff must guard against this.
A senior researcher develops and tests models: hybrid and process-based approaches for climate change, mixed-effects and Bayesian methods for calibration, models for mixed and uneven-aged forests, and the integration of remote sensing and laser scanning data into growth projections. The researcher also curates long-term experiments and plot networks, reviews model applications in carbon and policy contexts, and teaches the statistical and ecological foundations to students.
This manual gives no coefficients for any real yield table, SDI maximum, crown-area equation or growth model; the numbers in worked examples are illustrative. It does not cover the statistics of non-linear mixed-effects fitting in detail, Bayesian calibration, matrix (transition) models for uneven-aged stands, or the internal equations of named simulators. The relative-density thresholds (25%, 35%, 60%) are the commonly cited values of Long (1985) and vary by species and author. Harvest-scheduling formulations are introduced only to show how growth models feed them; they are treated in forest-management-yield-regulation. Measurement methods are in forest-mensuration and sampling design in forest-inventory-sampling.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
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