Terminology
Stand Density and Stocking
Silviculturists are often interested in three related measures of stand density: absolute density, relative density and
stocking. These measures can be used to describe a stand relative to some standard of comparison or to some condition
that meets a silvicultural objective. Silvicultural decisions are often based on such measures of stand density, and the
desired condition of the stand after treatment is usually described by these measures. Although the terms absolute density,
relative density and stocking have not always been used consistently, some general conventions and definitions have been
established (Walker, 1956; Bickford et al., 1957; Gingrich, 1964; Husch et al., 1982; Ernst and Knapp, 1985).
Absolute density (or simply density in common usage) is a quantitative, objective measure of one or more
physical characteristics of a forest stand expressed per unit area. Measures of absolute density are expressed quantitatively
as a tree count, area, volume or mass. Ecologists usually use the term density to refer exclusively to the number of
individuals per unit area. In forestry, however, the term can refer to any of several measures of site occupancy, including
number of trees, basal area or volume per unit area. Measures of density are usually restricted to trees larger than some
minimum size, usually expressed as a minimum dbh. Specifying this minimum size is important because absolute density
usually differs with differences in the minimum measured tree size.
Measures of relative density provide additional information by comparing an absolute density to a reference
value. An example of a measure of relative density is the ratio of the number of trees of a given species per acre to the
total number of trees per acre. Expressed as a percentage, this value has long been used by ecologists to define the relative
density of a species within a specified area.
Forest regeneration and growth can be greatly affected by relative stand density. Consequently, silviculturists
have developed various methods of expressing relative density. Virtually all silvicultural definitions of relative density
involve ratios. For example, Reineke’s (1933) stand density index provides a reference line describing the maximum
number of trees per acre for stands of a given mean dbh (Fig. 6.1). The maximum number of trees decreases rapidly as the
mean stand diameter increases. For any given stand, the observed number of trees and mean dbh can be used to compute
the ratio (or percentage) of the number of the maximum (reference) number of trees indicated by the stand density index
relation (Reineke, 1933; Schnur, 1937). Other common measures of relative oak stand density are based on tree–area
ratios or stocking percent (Chisman and Schumacher, 1940; Gingrich 1967; Ernst and Knapp, 1985).
In their application to oak forest types in North America, most measures of relative density are designed to
compare one or more absolute measures of stand density to a standard. The standard is often based on an observed
maximum absolute density for undisturbed natural stands at a comparable stage of development, but it may be based on
other limits or reference conditions. For example, crown competition factor (CCF) estimates stand density relative to a
minimum tree crown area per acre below which trees do not fully utilize available growing space (Krajicek et al., 1961).
The method of French ‘normes’ compares the observed number of trees and the mean height of dominant trees to both a
maximum density and the minimum number of trees necessary to maintain dbh growth below 2 mm (0.08 inch) per year,
which by European standards is considered most desirable for veneer production (Oswald, 1982).
Other measures of relative density that generally have not been applied to oak forests, but could be, include:
Curtis’s (1982) relative density index (references observed basal area per acre to that of an undisturbed stand with the
same quadratic mean diameter); Wilson’s (1946) relative spacing index (references the observed number of trees per acre
to the number of trees in an undisturbed stand having the same dominant height); and Drew and Flewelling’s (1977)
relative density index (references number of trees per unit area to the volume of the average tree). The latter method is
analogous to the graphical format for expressing the -3/2 power rule discussed earlier. Comprehensive reviews of
measures of relative density include those by Curtis (1970) and Stout and Larson (1988).
Stocking is a subjective term used to describe the adequacy of any observed level of stand density with respect
to a silvicultural goal (Bickford et al., 1957; Gingrich, 1964). The terms overstocked, understocked and fully stocked are
used to describe stocking adequacy relative to a specified silvicultural goal. Accordingly, a stand may be overstocked (too
dense) for one silvicultural objective and fully (i.e. appropriately) stocked for another, or may be overstocked at one age
and understocked at another.
In contrast to the term stocking, the term stocking per cent is a measure of relative density specifically
associated with the Gingrich-style stocking diagram. This diagram combines measures of absolute and relative density
into a single graphical format (Gingrich, 1967). Stocking per cent is a widely used measure of stand density in North
American oak silviculture. It is based on the relation between tree size and associated growing space requirements
discussed later in this chapter. The word stocking is often used incorrectly to refer to stocking per cent (a measure of
relative density). This sometimes creates confusion because, as discussed later, full stocking is synonymous with complete
utilization of growing space, which covers a wide range of stocking percentages on the Gingrich stocking diagram.
Normal stocking is a term used to describe undisturbed even-aged stands that are at or near maximum density
for their age. Normally stocked stands are characterized by a lack of gaps in the forest canopy and a relatively uniform
spacing between stems. Basal area and cubic foot volume are at or near their maximum for a given stand age and site
quality. Normally stocked stands (sometimes simply called normal stands) usually are identified subjectively based on
these criteria. Observations of the number, basal area and volume of trees per acre in normally stocked stands across a
wide range of stand age and site quality classes have been used to develop normal yield tables These tables specify the
expected maximum basal area and maximum cubic foot volume for unmanaged stands of a given age and site class. In
addition to their application to yield estimation, the tabulated values can be used as reference conditions to estimate the
relative density of other stands.
Maximum and minimum growing space
There are limits to the amount of growing space a tree of a given bole diameter can occupy. Although this may seem selfevident, the concept is central to quantifying stand density and stocking per cent in oak stands.
The actual amount of space that a tree occupies is difficult to measure because it includes crowns and roots that
overlap in three dimensions with other trees. Fortunately, for many silvicultural purposes, a tree’s growing space can be
adequately estimated as a circular area, or tree area, representing the crown. In this context, tree area is interpreted
geometrically as a tree’s area of influence or potential influence concentric to the tree bole; it is also highly correlated with
dbh. Estimates of the maximum area that a tree of a given dbh can occupy are usually developed from crown and dbh
measurements of open-grown trees. In contrast, estimates of the minimum area that a tree requires are usually developed
from measurements of tree diameters in normally stocked stands.
Trees that are open-grown throughout their lives develop the largest crowns possible for their dbh and species.
Consequently, open-grown trees have often been used to estimate the maximum area a tree of given species and dbh can
occupy. There is a high correlation between bole diameter and crown area of open-grown trees. This relation has led to the
development of equations for estimating the crown areas of open-grown trees from dbh for various oaks and associated
species in several regions in the eastern United States. The results have shown that the relation between maximum crown
width and bole diameter is often linear or nearly linear (Krajicek et al., 1961; Krajicek, 1967; Ek, 1974). An example is
the maximum crown width equation applicable to oaks and hickories in the Central Hardwood Region, which is given by:
CWmax = 3.12 + 1.829D
[6.9]
where CWmax is the estimated crown width (ft) of an open-grown upland oak or hickory, and D is tree dbh (inches)
(Krajicek et al., 1961). Assuming tree crowns are circular, squaring both sides of Equation 6.9 and multiplying by Π/4
defines maximum crown area (CAmax) in relation to dbh so that:
2
CAmax = 7.645 + 8.965D + 2.627D
[6.10]
2
CAmax therefore is the approximate circular crown area (ft in vertical projection) of an open-grown upland oak or hickory.
Maximum crown width equations also have been derived for other species and regions (Table 6.1). An exponent in the
diameter term of some equations indicates non-linearity in the relation. As in the derivation of Equation 6.10, equations in
Table 6.1 can be similarly expressed as crown area. Graphical presentation of equations facilitates comparisons among
species. For example, open-grown black walnut trees have larger crowns than oaks and hickories for a given diameter,
whereas shortleaf pines have smaller crowns. The maximum crown width of sugar maple may be larger or smaller than
that of oaks and hickories, depending on dbh (Fig. 6.7).
Assuming that maximum crown width equations adequately express the maximum amount of above-ground
growing space that a tree of a given diameter can occupy, we can estimate the fewest trees of a given dbh required to
completely occupy an acre, i.e. 43,560/CAmax. Alternatively, the maximum tree area for all the trees on any acre can be
calculated by summing their individual maximum crown areas (Equation 6.10 and Table 6.1). When the sum of the
2
maximum crown areas equals the area of an acre (43,560 ft ), the stand is said to have a maximum tree–area ratio
(TARmax) of 100%. This represents the condition where tree–area satisfies the minimum requirements for full utilization of
growing space given that tree crowns are, for their dbh, maximally extended. Maximum tree–area ratio (TARmax) therefore
is a relative measure of stand density that defines the percentage of an area (e.g. an acre) that would be utilized by trees
when all tree crowns are fully extended. When TARmax is less than 100%, the trees present would not utilize the available
growing space even when their crowns are fully extended. Consequently, reducing TARmax below 100% by thinning will,
at least temporarily, result in unutilized growing space. Calculating TARmax can be simplified by dividing equations for
open-grown crown areas (e.g. Equation 6.10, or Table 6.1) by 435.6, the area comprising 1% of an acre. For Equation
6.10, TARmax is given by:
TARmax = 0.0175 + 0.0205D + 0.00603D2
[6.11]
where TARmax is the maximum percentage of an acre that a tree of a given dbh (D) can occupy. The sum of TARmax for all
the trees on an acre is sometimes referred to as crown competition factor (CCF) (Krajicek et al., 1961) and is calculated
by summing TARmax for individual trees as follows:
2
CCF = Σ(0.0175 + 0.0205Di + 0.00603Di )
= 0.0175N + 0.0205ΣDi + 0.00603Σ Di
2
[6.12]
[6.13]
Table 6.1. Equations for estimating open-grown crown widths from dbh for oaks and some
commonly associated species.
---------------------------------------------------------------------------------------------------------------------------Maximum crown
a
Species (location)
width
Source
---------------------------------------------------------------------------------------------------------------------------American elm (Wisconsin)
2.829 + 3.456D0.8575
Ek, 1974
American basswood (Wisconsin)
0.135 + 3.703D0.7307
Ek, 1974
Black cherry (Wisconsin)
0.621 + 7.059D0.5441
Ek, 1974
Black oak (Wisconsin)
4.504 + 2.417D
Ek, 1974
Black walnut (unspecified)
4.873 + 1.993D
Krajicek, 1967
Black walnut (Wisconsin)
4.901 + 2.480D
Ek, 1974
Bur oak (Wisconsin)
0.942 + 3.539D0.7952
Ek, 1974
0.7381
Green ash (Wisconsin)
4.755D
Ek, 1974
Jack pine (Quebec)
2.036 + 1.736
Vezina, 1963
Loblolly pine (unspecified)
4.78 + 1.56D
Roberts and Ross, 1965
Northern red oak (Wisconsin)
2.850 + 3.782D0.7968
Ek, 1974
Oaks and hickories (Iowa)
3.12 + 1.829D
Krajicek et al., 1961
Pin oak (unspecified)
9.06 + 1.525D
Krajicek, 1967
Red maple (Wisconsin)
4.776D0.7656
Ek, 1974
Shagbark hickory (Wisconsin)
2.369 + 3.548D0.7986
Ek, 1974
Shortleaf pine (Missouri)
2.852 + 1.529D
Rogers, 1983
Sugar maple (Wisconsin)
0.868 + 4.150D0.7514
Ek, 1974
Sugar maple (Eastern US)
12.08 + 1.32D
Smith and Gibbs, 1970
Sweetgum (unspecified)
2.65 + 1.975D
Krajicek, 1967
White oak (Wisconsin)
3.689 + 1.838D
Ek, 1974
---------------------------------------------------------------------------------------------------------------------------------------a
Crown width in feet given tree dbh (D) in inches; corresponds to CWmax in text. Assuming tree crowns are circular in crosssection, maximum crown area in square feet is equal to (CWmax)2•/4.
2
where summations (Σ) are over all trees per acre, Di is the dbh of tree i and N is number of trees per acre. Note that Σ Di
is equal to the stand basal area in square feet per unit area divided by Σ/576. A CCF of 100 (or equivalently, ΣTARmax =
100%) therefore is usually interpreted as the approximate lowest density at which a stand fully utilizes above-ground
growing space. Stands with CCFs below 100 are certain to have canopy gaps. CCF values near 200 have been observed
for undisturbed oak–hickory stands (Krajicek et al., 1961).
Just as trees have a maximum area they can occupy, they also have a minimum tree area that is necessary for
good physiological function and survival. However, minimum tree area is derived quite differently from its maximum
tree-area counterpart. Unlike maximum tree area, which can be estimated from open-grown trees, minimum tree area is
difficult to observe directly for individual trees. Minimum tree area requirements nevertheless can be estimated from data
obtained from undisturbed, normally stocked, even-aged stands. Estimation is based on deriving minimum tree–area ratio
(TARmin) equations that express tree growing space requirements for normally stocked stands (Chisman and Schumacher,
1940). Like TARmax, TARmin expresses tree area in percent of an acre.
Just as maximum tree area is a linear function of diameter and diameter squared (Equations 6.9 and 6.10), a
tree’s minimum tree area can be similarly expressed by:
2
TARmin = c0 + c1D + c2D
[6.14]
where TARmin is the estimated minimum per cent of an acre required by a tree of a given dbh (D) in a normally stocked
forest. Unlike the maximum tree–area coefficients, the coefficients for the minimum tree–area equation are not derived
from measurements of crown diameters. Instead, they are estimated by regression by assuming the sum of the tree areas
2
for all trees on an acre of undisturbed, normally stocked forest is equal to 43,650 ft , or 100% of an acre (Chisman and
Schumacher, 1940; Gingrich, 1967).
Fig. 6.7. Estimated open-grown crown widths of oaks and hickories and three commonly associated species in relation to
bole diameter (dbh). See also Table 6.1. (Adapted from Ek, 1974 (sugar maple), Krajicek, 1967 (black walnut), Krajicek et
al., 1961 (oak–hickory), Rogers, 1983b (shortleaf pine).)
Minimum tree area then can be expressed directly as a percentage of an acre. This expression has been termed
stocking per cent (S%) (Gingrich, 1967) and is given by:
2
S% =Σ(b0 + b1Di + b2D2 )
= b0N + bΣDi + b2ΣD2
2
[6.15]
[6.16]
where summations (Σ) are over all trees per acre, S% is the percentage of an acre filled by the minimum tree areas of all
trees on that acre, Di is the dbh of tree i, N is the number of trees per acre, and b0, b1 and b2 are coefficients (usually
estimated by regression). The stocking percentage represented by a single tree can be derived by solving Equation 6.16 for
N = 1.
When stocking percentage equations are expressed on a per acre basis (e.g. Equation 6.15 or 6.16), equations for
minimum tree area in square feet can be derived by multiplying each term in the equation by 435.6 (the number of square
feet in 1% of an acre). Although stocking percentage is usually the relative density measure of choice, rescaling to square
feet facilitates comparing the estimated tree areas representing maximum and minimum growing space (Fig. 6.8). Other
factors being equal, the closer a tree’s crown is to its maximum size for its dbh, the faster the tree’s diameter and gross
volume growth.
The minimum tree–area ratio is reportedly independent of stand age and site quality (Chisman and Schumacher,
1940; Gingrich, 1967), and can be applied to mixed as well as to pure stands. The methodology for deriving minimum
tree–area equations is described in more detail by others (Chisman and Schumacher, 1940; Gingrich, 1967; Roach, 1977;
Ernst and Knapp, 1985; Stout and Nyland, 1986).
Fig. 6.8. Estimated maximum and minimum tree areas in relation to bole diameter (dbh) for upland oaks and hickories in the
Central Hardwood Region. The area between the two lines represents the approximate biological range of crown areas for
individual trees. (From Gingrich, 1967.)
Oak and associated forest types are seldom comprised of a single species. In applying stocking equations, it is
therefore important to recognize differences in tree–area ratios among species. In developing stocking equations,
coefficients for individual species can be derived by incorporating species-specific terms into stocking equations so that
the Equation 6.15 expands to the more general form:
no. spp.
no.trees
no.trees
2
S% =Σ (b0jNj + b1jΣ Dij + b2jΣDi )
j=1
i=1
i=1
[6.17]
where the outer summation (Σ) is over all species; the inner summations are over all i trees of species j; b0j, b1j and b2j are
coefficients specific to species j; Nj is the number of trees of species j; and Dij is the diameter of tree i of species j (Roach,
1977).
In forests such as the oak–hickory type of the Central Hardwood Region, the minimum tree–area ratios of the
predominant species do not differ significantly (Krajicek et al., 1961; Gingrich, 1967). A single set of coefficients
therefore can be used to represent the major species of the forest type. In other forest types, tree–area ratios differ
appreciably among species (Roach, 1977; Stout and Nyland, 1986; Zhang et al.,1995). When such differences occur,
separate coefficients for individual species or species groups can improve the accuracy of relative density equations. This
is the case in the Allegheny hardwood forests of Pennsylvania, which are comprised of mixed stands of black cherry,
yellow-poplar, red maple, white ash, sugar maple, black birch, yellow birch, American beech, oaks and other species.
Analysis of species-specific tree–area ratios identified three species groups with significantly different tree–area ratios
(Stout and Nyland, 1986; Zhang et al., 1995). Stocking equations based on tree–area ratios also have been derived for
northern red oak and various forest types of the eastern United States that often include oaks (Table 6.2).
The silvicultural value of tree–area ratios and stocking percentage for defining relative density is reinforced by
their demonstrated independence of stand age and site quality (Chisman and Schumacher, 1940; Gingrich, 1967). They
also have been shown to be little influenced by variation in stand structure (Gingrich, 1967). Minimum tree–area ratio
equations and equivalent stocking percentage equations can be used to calculate an approximate average upper limit of
stand density corresponding to that for normally stocked stands. This upper limit, which can be graphically expressed as a
line on a stand density diagram, has been termed average maximum density or average maximum competition (Ernst and
Knapp, 1985).This line is generally interpreted as the level of density that stands tend to return to in the absence of
disturbance (Gingrich, 1967; Ernst and Knapp, 1985). Nevertheless, stand density is likely to fluctuate around this line
due to variation in weather, minor outbreaks of insects and disease, and other factors associated with ‘regular’ tree
mortality. A potential deficiency of all measures of maximum relative density is the necessarily subjective selection of
stands or plots used to derive tree–area ratios.
References
Beck, D.E. (1986) Thinning Appalachian pole and small sawtimber stands. Society of American Foresters Publication 86-02, pp. 85–98.
Bickford, C.A., Baker, F.S. and Wilson, F.G. (1957) Stocking, normality, and measurement of stand density. Journal of Forestry 55, 99–
104.
Canadell, J. and Rodà, F. (1991) Root biomass of (Quercus ilex) in a montane Mediterranean forest.Canadian Journal of Forest Research
21, 1771–1778.
Chisman, H.H. and Schumacher, F.X. (1940) On the tree–area ratio and certain of its applications.Journal of Forestry 38, 311–317.
Curtis, R.O. (1970) Stand density measures: an interpretation. Forest Science 16, 403–414.
Curtis, R.O. (1982) A simple index of stand density for Douglas-fir. Forest Science 28, 92–94.
Dale, M.E. (1972) Growth and yield predictions for upland oak stands 10 years after initial thinning.USDA Forest Service Research
Paper NE NE-241.
Drew, J.T. and Flewelling, J.W. (1977) Some recent Japanese theories of yield–density relationships and their application to Monterey
pine plantations. Forest Science 23, 517–534.
Ek, A.R. (1974) Dimensional relationships of forest and open grown trees in Wisconsin. University of Wisconsin Forestry Research Note
July.
Ernst, R.L. and Knapp, W.H. (1985) Forest stand density and stocking: concepts, terms, and the use of stocking guides. USDA Forest
Service General Technical Report WO WO-44.
Frothingham, E.H. (1912) Second-growth hardwoods in Connecticut. USDA Forest Service Bulletin 96.
Gevorkiantz, S.R. and Scholz, H.F. (1948) Timber yields and possible returns from the mixed-oak farmwoods of southwestern
Wisconsin. USDA Forest Service Lake States Forest Experiment Station Publication 521.
Gingrich, S.F. (1964) Criteria for measuring stocking in forest stands. Proceedings of 1964 Society American Foresters and National
Convention, pp. 198–201.
Gingrich, S.F. (1967) Measuring and evaluating stocking and stand density in upland hardwood forests in the Central States. Forest
Science 13, 38–53.
Gingrich, S.F. (1971) Management of young and intermediate stands of upland hardwoods. USDA Forest Service Research Paper NE
NE-195.
Givnish, T.J. (1986) Biomechanical constraints on self-thinning in plant populations. Journal of Theoretical Biology 119, 139–146.
Goelz, J.C.G. (1991) Generation of a new type of stocking guide that reflects stand growth. USDA Forest Service General Technical
Report SE SE-70, Vol. 1, pp. 240–247.
Goelz, J.C.G. (1995) A stocking guide for southern bottomland hardwoods. Southern Journal of Applied Forestry 19, 103–104.
Harper, J.L. (1977) Population Biology of Plants. Academic Press, London.
Husch, B., Miller, C.I. and Beers, T.W. (1982) Forest Mensuration. Wiley, New York.
Hutchings, M. (1983) Ecology’s law in search of a theory. New Scientist 16, 765–767.
Kershaw, J.A., Jr and Fischer, B.C. (1991) A stand density management diagram for sawtimber-sized mixed upland central hardwoods.
USDA Forest Service General Technical Report NE NE-148, pp. 414–428.
Krajicek, J.E. (1967) Maximum use of minimum acres. Proceedings 9th Southern Forest Tree Improvement Conference, pp. 35–37.
Krajicek, J.E., Brinkman, K.A. and Gingrich, S.F. (1961) Crown competition – a measure of density.Forest Science 7, 35–42.
Leak, W.B. (1981) Do stocking guides in the Eastern United States relate to stand growth? Journal of Forestry 79, 661–664.
Leary, R.A. and Stanfield, D. (1986) Stocking guides made dynamic. Northern Journal of AppliedForestry 3, 139–142.
Lonsdale, W.M. (1990) The self-thinning rule: dead or alive? Ecology 71, 1373–1388.
McGill, D., Martin, J., Rogers, R. and Johnson, P.S. (1991) New stocking charts for northern red oak. University of Wisconsin Forestry
Research Notes 277.
McGill, D.W., Rogers, R., Martin, A.J. and Johnson, P.S. (1999) Measuring stocking in northern red oak stands in Wisconsin. Northern
Journal of Applied Forestry 16, 144–150.
McMahon, T. (1973) Size and shape in biology. Science 179, 1201–1204.
McMahon, T.A. and Bonner, J.T. (1983) On Size and Life. Scientific American Books, New York.
Miyanishi, K., Hoy, A.R. and Cavers, P.B. (1979) A generalized law of self-thinning in plant populations. Journal of Theoretical Biology
78, 439–442.
Norberg, R.A. (1988) Theory of growth geometry of plants and self-thinning of plant populations: geometric similarity, elastic similarity,
and different growth modes of plant parts. American Naturalist 131, 220–256.
Nowak, C.A. (1996) Wood volume increment in thinned, 50- to 55-year-old, mixed-species Allegheny hardwoods. Canadian Journal of
Forest Research 26, 819–835.
Oswald, H. (1982) Silviculture of oak and beech high forests in France. Proceedings of Broadleaves in Britain, Future Management &
Research Symposium, pp. 31–39.
Putnam, J.A., Furnival, G.M. and McKnight, J.S. (1960) Management and inventory of southern hardwoods. USDA Forest Service
Agriculture Handbook 181.
Reineke, L.H. (1933) Perfecting a stand-density index for even-aged forests. Journal of Agricultural Research 46, 627–638.
Roach, B.A. (1977) A stocking guide for Allegheny hardwoods and its use in controlling intermediate cuttings. USDA Forest Service
Research Paper NE NE-373.
Roberts, E.G. and Ross, R.D. (1965) Crown area of free-growing loblolly pine and its apparent independence of age and site. Journal of
Forestry 63, 462–463.
Rogers, R. (1983) Guides for thinning shortleaf pine. USDA Forest Service General Technical Report SE SE-24, pp. 217–225.
Sampson, T.L. (1983) A stocking guide for northern red oak in New England. MS thesis, University of New Hampshire, Durham.
Sampson, T.L., Barrett, J.P. and Leak, W.B. (1983) A stocking chart for northern red oak in New England. University of New Hampshire
Agricultural Experiment Station Research Report 100.
Schnur, G.L. (1937). Yield, stand, and volume tables for even-aged upland oak forests. USDA Technical Bulletin 560.
Smith, H.C. and Gibbs, C.B. (1970) A guide to sugarbush stocking. USDA Forest Service Research Paper NE NE-171.
Sonderman, D.L. (1985) Stand density – a factor affecting stem quality of young hardwoods. USDA Forest Service Research Paper NE
NE-561.
Sorrensen-Cothern, K.A., Ford, E.D. and Sprugel, D.G. (1993) A model of competition incorporating plasticity through modular foliage
and crown development. Ecological Monographs 63, 277–304.
Sprugel, D.G. (1984) Density, biomass, productivity, and nutrient-cycling changes during stand development in wave-regenerated balsam
fir forests. Ecological Monographs 54, 165–186.
Stout, S.L. (1991) Stand density, stand structure, and species composition in transition oak stands of northwestern Pennsylvania. USDA
Forest Service General Technical Report NE NE-148,pp. 194–206.
Stout, S.L. and Larson, B.C. (1988) Relative stand density: why do we need to know? USDA ForestService General Technical Report
INT INT-243, pp. 73–79.
Stout, S.L. and Nyland, R.D. (1986) Role of species composition in relative density measurement in Allegheny hardwoods. Canadian
Journal of Forest Research 16, 574–579.
Trimble, G.R. Jr (1968) Multiple stems and single stems of red oak give same site index. Journal of Forestry 66, 198.
Vezina, P.E. (1963) More about the crown competition factor. Forestry Chronicle 39, 313–317.
Walker, N. (1956) Growing stock volumes in unmanaged and managed forests. Journal of Forestry 54,378–383.
Weller, D.E. (1987a) A reevaluation of the _3/2 power rule of plant self-thinning. EcologicalMonographs 57, 23–43.
Weller, D.E. (1987b) Self-thinning exponent correlated with allometric measures of plant geometry. Ecology 68, 813–821.
Weller, D.E. (1989) The interspecific size–density relationship among crowded plant stands and its implications for the _3/2 power rule
of self-thinning. American Naturalist 133, 20–41.
Westoby, M. (1984) The self-thinning rule. Advances in Ecological Research 14, 167–225.
White, J. (1981) The allometric interpretation of the self-thinning rule. Journal of Theoretical Biology89, 475–500.
White, J. (1985) The thinning rule and its application to mixtures of plant populations. In: Studies on Plant Demography. Academic
Press, New York, pp. 291–309.
White, J. and Harper, J.L. (1970) Correlated changes in plant size and number in plant populations. Journal of Ecology 58, 467–485.
Wilson, F.G. (1946) Numerical stocking in terms of height. Journal of Forestry 44, 758–761.
Yoda, K., Kira, T., Ogawa, H. and Hozumi, K. (1963) Self-thinning in overcrowded pure stands under cultivated and natural conditions.
Journal of Biology 14, 107–129.
Zeide, B. (1985) Production of unmanaged bottomland hardwoods in Arkansas. Proceedings of the Central Hardwood Forest Conference
V. University of Illinois, Urbana-Champaign, pp. 118–124.
Zeide, B. (1987) Analysis of the 3/2 power law of self-thinning. Forest Science 33, 517–537.
Zeide, B. (2001) Thinning and growth: a full turnaround. Journal of Forestry 99(1), 20–24.
Zhang, L., Oswald, B.P., Green, T.H. and Stout, S.L. (1995) Relative density measurement and species composition in the mixed upland
hardwood forests of North Alabama. USDA Forest Service General Technical Report SRS SRS-1, pp. 467–472.