Stand Density and Stocking Terminology 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 self- evident, 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 maximum crown areas equals the area of an acre (43,560 ft2), 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 ) [6.12]

2 = 0.0175N + 0.0205ΣDi + 0.00603Σ Di [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 cross- section, 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 for all trees on an acre of undisturbed, normally stocked forest is equal to 43,650 ft2, 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 ) [6.15]

2 = b0N + bΣDi + b2ΣD2 [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. 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