Stand Bole Volume ©

In estimating stand volume, it is essential to realise that:

• the estimate is based on a sample
• the intensity of sampling on an area basis may be very low, even very much less than1%
• within the area selected as the sample, one subsamples further both in selecting the trees to sample for volume and in selecting the points of measurement on the individual sample trees.

Methods of estimating stand volume

• Ground Based Methods
  • Sample tree
  • Tree volume tables
    • 1-way
    • 2-way
    • 3 or 4 way
  • Stand volume tables
  • Yield tables
• Aerial photograph based methods
  • Area tree volume tables
  • Areial stand volume tables
  • Visual comparisons
  • Combined aerial and ground (truthing) methods

Sample tree method

The underlying theory is to isolate the tree of mean volume in the stand and carefully measure its volume, then multiply by the number of trees in the stand. An alternative approach is to calculate the volume per unit of basal area of the mean tree and multiply this value by G to derive stand volume.

In principle, the method is simple and direct. In practice, difficulty is encountered in finding the tree of mean volume. Sampling is necessary. If this sampling is done objectively, a number of trees will be required to give any chance of securing a reasonable estimate of mean tree volume. On the other hand, subjective sampling may give as good a result with many fewer sample trees. The ultimate is to sample one tree only.

Various clues are available to aid selection of the tree of mean volume, namely:

1. mean g
2. mean h
3. information on taper and bark thickness.

It is pointless spending too much time obtaining refined estimates of mean g, mean h, mean taper, etc. of the stand because, for mathematical reasons, the tree with these characteristics, even if it does exist, will not have mean tree volume, the reason being that the product of means is not equal to the mean of products. Thus, the method has a fundamental limitation.

As the tree of exact mean volume is unlikely to exist physically in the stand, it is more appropriate to select two or three or more trees of dimensions comparable with the mean dimensions of trees in the stand, measure their volumes directly, and take the mean. However, the more trees taken, the more the essential simplicity of the method is lost.

The simple arithmetic mean tree method is still used sometimes in resource inventory and preliminary working plan inventory of even-aged stands. It is useful in extension forestry for quick, cheap estimates of stand volume.

One variation of the arithmetic mean tree method is to apply it independently to groups or strata in the stand based on diameter class or basal area class. The strata should be set up to contain an equal number of trees or an equal total basal area. Stand volume is then obtained by summing the class volumes.

Tree volume tables

The solution to this requirement lies in establishing relationships between the volumes of trees or sections and easily measured tree dimensions. Estimation of the volume of a tree or section then consists of measuring the appropriate dimensions and reading the corresponding volume from a graph, table or formula.

A tree volume table is thus a statement of the expected volume of a tree of particular dimensions in a particular stand or population. The application of the table depends on the variables embodied in it, e.g. a table based on d only will apply to a specific stand; whereas one based on d, h, bark thickness and an expression of taper may apply to a range of species over a range of conditions if these variables adequately account for the variation in tree volume.

The essential purpose of a tree volume table is to estimate the volume of standing trees. A volume table reduces the tedious task of measuring individual tree volumes to the relatively simple task of measuring a few dimensions on the trees encountered.

Tables may either be direct reading or indirect reading. Direct reading tables are presented most often in tabular form but tables presented as graphs or alignment charts (see later) are not uncommon. The volume equation is the mathematical expression of the indirect reading volume table, e.g.,

  V = a + b.g + c.h+ d.g.h.

Volume tables are compiled by graphical or mathematical analysis of sample tree data. The common dependent variables are total volume, merchantable volume and pruned section volume and the common independent variables are d, total or merchantable h, an expression of taper (e.g. dub1.5 m - dub4.5 m), and an expression of bark thickness at approximately breast height.

Graphical and tabular volume tables are referred to as 1-way, 2-way, etc., depending on the number of independent variables used. Use of the terms local, standard, regional, general, and universal should be discouraged because the scope of application, and the number of independent variables, are not necessarily connected: it all depends on how well those variables embodied in the table account for variation in tree volume.

Features of a good volume table

• Name of species concerned
• Locality and type of stand represented
• Date of preparation
• Author
• Range and frequency of basic data by d and h classes
• Method of choosing sample trees
• Method of taking sectional measurements
• Method of computing volume
• Volume units used
• Nature of any limits of merchantability for stump or top diameters
• Method of construction
• Accuracy and precision of model
• Trends in residuals, i.e. unexplained variation
• Result of test with new set of data
• The table itself and/or formula.

1-Way Tree Volume Tables

Three types of 1-way table are common in even-aged stands, viz. volume curve, volume line and tariffs.

Volume Curve

The relationship is established based on sample trees selected on a stratified random basis, the curve of best fit being located by eye or calculated e.g.

  Log v  = a + b d             )
      v  = a + b log d         ) Equations 
      v  = a + b d + c d 2     ) worth
      v  = a d b               ) testing	
  Log v  = a + b log d         )

Volume line (= v /g line)

  v = a + b g.

The main advantage of the volume line over the volume curve is that the line is easier to fit by eye and fewer observations are needed to establish it.

The regression is best calculated by the method of least squares. Fitting the line by eye may introduce bias depending on the variability of the data and the experience of the operator.In practice, the volume line for a stand is established from sample trees. Although objective selection of the sample is desirable to avoid bias, subjective selection is widely adopted in practice. For example, the operator, after looking at the variation in each diameter class, proceeds to select average trees from each class. This reduces the size of sample to a minimum. Bias can be kept under reasonable control if the operator is familiar with the population.

Hooke (1963) and Demaerschalk and Kozak (1974) suggest that when the relationship is known to be linear, the best estimates of the slope of the regression line are obtained when half of the observations are taken at each end of the range of the independent variable. When there is any doubt about linearity, the observations should be distributed over the whole range either uniformly or with a greater concentration of observations towards the ends of the range.

The normal procedure in processing the data is to plot the values in the field as a check on the assumption of linearity, atypical sample trees, and errors, and calculate the regression later in the office.

The volume line for a particular stand is a 1-way volume table for that stand at that time. The volume of any tree in the stand can be calculated by substituting its g in the equation or reading from the graph. Stand volume is derived by summing the individual tree volumes or substituting in the formula:

      V = N*a + b* (sum of all g)
							
where V	= stand volume
      N 	= number of trees in the stand 
     (sum of all g)	= stand baob (= G)
      and a and b are the regression constants.
											
The above equation is derived as follows:
    Tree 1         v1	= a + b g1
    Tree 2         v2	= a + b g2
		   ..            ..     ..
		   ..            ..     ..
    Tree n         vn = a + b gn
               ___________________  
  Thus        Sum Vol = N*a + b*(sum of g)

Tariffs

A tariff comprises a series of volume curves or lines relating diameter at breast height, tariff number and volume. Knowing the first two, one can read off volume.

The great value of tariffs, if the tariff applicable to a particular stand at a particular time can be determined, is that there is little or no need for sample trees to compile the appropriate volume curve or line. Consequently methods of nominating tariffs are of considerable importance.

Method 1:

• Determine the tariff directly, nominating it for a stand at a particular time with a minimum number of sample trees.
• Jolly (South Australia) found for P. radiata over a wide range of age and stand conditions that volume lines of volume u.b to 10 cm converged on the abscissa at a g of 0.008 m2. This led to the method of selecting a number of sample trees of approximately mean volume and joining the plotted position of their mean to 0.008 m2, which is equivalent to nominating one of a series of tariffs radiating from the 0.008 value.
• Note: Under the Imperial System, the individual volume lines in Jolly's Tariffs were referenced by the merchantable volume u.b. (to 4" d.u.b.) at a g (overbark) of 0.5 ft2, e.g. Tariff No.15 indicated a volume line for which a tree of 0.5 ft2 g would have a merchantable volume of 15 ft3. The most sensible adjustment in converting to the metric system would be to reference the volume lines to tree volume (m3) at a g of 0.05 m2.
• Hummel (1955) in the U.K. found that the volume lines converged at approx. the 0.003 m2 point. This led to the production of general tariff tables, i.e. a series of lines radiating from 0.003 m2 through successive values of volume with a defined volume interval between them at a g of 0.09 m2.
• The tariffs are identified by numbers equivalent to these values of volume, e.g. Tariff 0.1, 0.2, etc.
• The BFC tariffs have been used widely to estimate the volume of thinnings, the volume line to apply at a given case being nominated from the volumes of felled sample trees.

• Determine the tariff indirectly from standards
• Compile a series of stand height curves (h vs. d) and record for each curve the accompanying tariff by sampling to establish the v / g line.
• Determine the tariff for a given stand at a particular time by matching its actual stand height curve with one of those in the established series of curves.
• The rationale for this method is that it is much easier to derive the stand height curve than the volume line for sample trees. For an example of this method, see Turnbull and Hoyer (1965), Dept. Nat. Res. Manage. Report. No. 8, Olympia, Washington).

• Compile the tariffs for a range of stands directly and then relate the statistics of the tariffs ('a' and 'b' values) to stand variables.
• Predominant height (hdom) for example, has been found to be a reliable index of the statistics of the volume line (i.e. good correlations can often be established between the regression constants and coefficients and stand predominant height). This means that the tariff applicable to a stand can
• be nominated as soon as hdom is known (e.g. Finch, l957) .

i.e.,    a = k1 + k2 hdom                       (1)
 and     b = k3 + k4 hdom                       (2)

Substituting (1) and (2) in the formula 
         v = a + b g, we obtain:
         v = k1 + k2 hdom + k3 g + k4 hdom g     (3)

This method is now widely used in Australia and overseas for compiling tariffs.

V/g Relationship in Uneven Aged Stands

2-way Tree Volume Tables

The table is compiled from sample tree information, the number of sample trees depending on:

• variability of the material
• precision of estimate required
• method of compilation - more samples are needed if the relationships are fitted by hand, i.e. subjectively.

Now  v = g x h x F
 where 	v = tree volume (over or under bark)
  g  =  basal area (over or under bark)
  h  =  height
  F  =  form factor

If F is constant for a range of tree basal areas at a given height, then v varies linearly with g. If F varies in a regular way for a range of basal areas at a given height, then the volume/basal area relationship will be a simple curve. If F varies erratically, then the volume/basal area relationship will be complex.

In compiling a 2-way table, one seeks to detect the pattern of change of volume with change of basal area and height, and then to express it in the simplest and most efficient way.

Volume Table Compilation

Six phases of work are involved in preparing a table:

1. Defining and then sampling the population
2. Computing volume of sample trees
3. Sorting data by classes
4. Fitting a model
5. Testing adequacy of fit
6. Use of table (or function).

• Graphical method: This is the oldest method of volume table compilation and is still used in practice but since the advent of computers its use has waned in favour of mathematical analysis. Nevertheless, the method is still useful in the initial phase of volume table compilation to establish the types of relationships to examine in the mathematical analysis.
• Mathematical (regression) analysis
• Alignment chart methods: a graphical representation of a relationship or function in which the axes representing the variables are separate straight lines or curves. The axes do not intersect.The scales and relative position of the axes are so arranged (trial and error) that a connecting line across the chart intersects the axes at particular values which satisfy the function in question. A slide rule is a well-known example of an alignment chart. Alignment charts for estimating tree volume are common in forestry practice in the USA and Canada. They are not used to any great extent in Australia.

Indirect Approach to Volume Table Compilation

The tables are derived by plotting mean form factor (tree volume u.b. expressed as a proportion of the volume of a cylinder of diameter equivalent to tree dbhob and height equal to tree height) against dbhob. Then, Volume = g x h x F

Regression analysis

Analysis is based on the premise that volume is related to the chosen independent variables according to a definite mathematical function, which, one hopes, will reveal itself from a series of samples. With graphic techniques, this mathematical function is not necessarily defined explicitly but it is implicit in the method. The function is present, but its expression and solution are handled graphically. When least squares fitting is used, the form of the equation expressing the relation of volume to size measurements must be decided on beforehand. The model must be sound biologically and mathematically. The constants giving the best fit for the chosen equation are then calculated.

The use of weighting is necessary when variances are not homogeneous. Weights are usually made inversely proportional to the known or assumed variance in volume about the regression surface. Weighting results in more precise estimates of the coefficients. Advantages of Regression Analysis

• Solution of the expression is handled objectively - only one solution can result for the data and the chosen equation.
• The standard error of the estimate and the correlation coefficient can readily be calculated - these express the precision and reliability of the relationship.
• The number of sample trees needed as basic data is reduced.
• The approach is flexible. Additional data can be incorporated as they become available to improve the relationship between volume and the independent variables. Thus, precision and reliability of the table improve with time.

• Measurement of upper stem diameters is time consuming and expensive
• Variation in tree form has a much smaller impact on tree volume (or weight) than variation in d or h.
• With some species, form is relatively constant regardless of tree size.
• With other species, tree form is often correlated with tree size, so that the variables d and h often explain much of the variation in volume (or weight) caused by form differences.

1.* Constant form factor y = b1*d^2  			 
2.# Combined variable    y = b1 + b1*d^2*h                                        
3.# Australian equation or generalised combined variable
                         y = b0 + b1*d^2 + b2*h + b3*d^2*h
4.* Logarithmic          y = b1*d^(b2)*h^(b3)      		
5.# Generalised logarithmic	
                         y = b0 + b1*d^(b2)*h^(b3)
6.* Honer transformed variable
                         y = d^2 / (b0 + b1*h^(-1)) 
7.# Form class           y = b0 + b1*d^2*h*f		 
 
*	Generally used only for predicting total stem volume or weight.
#	More flexible - used for predicting total and merchantable stem volume or weight.
  y  =  some measure of stem content
		d  =  dbhob
		h  =  some measure of tree height
		f  =  an expression of tree form
	bo, b1, b2, b3   are  coefficients.

The most satisfactory regression is indicated by the form of the relationship between:

• h and the constants and coefficients of the regression of v against g (or d) within height classes;
• g (or d) and the constants and coefficients of the regression of v against h within g (or d) classes.

	a = a1 + b1h  and 	b = a2 + b2h
- but as the regression of v on g is frequently linear, we can write:
	v 	= a + b.g
		= a1 + b1 h + (a2 + b2 h) g
		= a1 + b1 h + a2 g + b2h g
where g is tree baob.

	v 	=   a + b. h g
  Thus  v /h g	=  a/h g + b
  i.e. 	F  	=  a/h g +  b  (equation to a hyperbola)

       i.e. v = a h g (which is the Form Factor Equation)

For volume tables derived by least squares solution, accuracy (or reliability) is indicated by the correlation coefficient and is best judged by a Chi-Squared-test on a carefully selected sample. Precision is best judged by the standard error of the estimate.

In recent years, high speed computers have increased enormously the scope of investigations into suitable equations. The usual procedure now is to start with a comparatively simple model such as the Australian Equation, and use a standard test to determine what functions of diameter and height not included in the trial model show a significant correlation with the dependent variable. These variables are then added to the regression. In this fashion, the best possible volume equation for the sample at hand is established. An example of this is the "Second Growth Blackbutt Model" compiled by the NSW Forestry Commission. The general form of this model is:

  vT  =  a +  b.g  +  c.h  +  d.gh  +  e.h 2  +  f.gh 2
where 	vT 	= volume underbark (m3) above a 1.2 m stump
       	h	= bole height (m) above ground
and    	g 	= baob (m2).

Under Australian conditions with plantation conifers, the Australian equation gives a satisfactory fit for total volume and merchantable volume to 10 cm and sometimes to 15 cm dub. The combined variable equation is almost as good.

Problems of curvilinearity arise if the merchantable limit is much greater than 10 cm or if the range of diameters (dbhob) is extensive. A wide diameter range was one problem Cromer, McIntyre and Lewis (1955) ran into with their General Volume Table for P. radiata. They overcame the problem by using three separate Australian equations to cover the full range of the data.

Incorporating the reciprocal of basal area in the Australian equation allows merchantable volumes to upper diameter limits (10, 12, 15 cm etc. top dub) to be computed more accurately (Spurr, 1952 ). This has been utilised by Vanclay and Shepherd (1983 - QFD Tech. Pap. No. 36) to develop volume functions for plantation species in Queensland, the model used being of the form:

  v  =  a + b. g  + c. hdom  +  d. g. hdom   + (e + f. hdom ) / (g + i)
where	    g  	=  tree BAOB
		hdom   	=  stand predominant height,
and a, b, c, d, e, f and i are constants.

Volume to various utilisation limits is sometimes derived from total volume by means of conversion factors (CFs). These factors increase with diameter and may be computed from an exponential function of the form given below. A separate solution is needed for each merchantable limit.

		CF  =  a + be^(cd)
where	d is dbh
		a, b and c are coefficients
and	e is 2.7183.

In recent times, volume (and weight) prediction equations have been devceloped which include the merchantability limit as a further independent variable (e.g. Bary and Borough 1980 , Clutter et al. 1983). With equations of this type, predicted volumes to various merchantability limits can be obtained using a single equation, e.g.:

  vt  =  ((a1 + a2 d 2 + a3 h + a4d  2h) a5 h ) / (h - 1.3)
then,	vm  	=  vt - (a6 + a7 h m 3.5) / d  1.5)
where	vt  	=  total stem volume (or weight) under bark from ground level to tip
  d  	=  dbh
  h   	=  total height
  vm  	=  merchantable volume (or weight) under bark to a specified under bark diameter limit
  m	=  merchantability limit given as under bark diameter, e.g. 10 cm 	     
  a1- a7	=  regression coefficients

Application of 2-way Tables

1. to measure all trees for dbhob (and derive a stand table) and sub-sample for heights. The height data from the sub-sample are used to establish a height/diameter regression relationship, i.e. the stand height curve.
2. read from the volume table the volume corresponding to the tree of average diameter (dbhob) and height for each class.
3. multiply tree volume by class frequency.
4. sum the class volumes.

3-Way and 4-Way Tables

• dub1.5m - dub4.5m
• dub1.5m - dub7.5m
• dub4.5m / dub1.5m
• or form quotient (i.e. d5/do)

Stand volume tables

Stand volume tables have been used in Europe for over a century, viz.:

Stand volume = G  x some expression of stand height x stand form factor.

Taking quite a different approach to European foresters, Spurr (1952) demonstrated for a wide range of stands (i.e. irrespective of species, species structure, age structure, site, or stocking) that regressions of stand volume on stand basal area within stand height classes were linear, as were the regressions of the regression constants and coefficients themselves on stand height. This suggested the suitability of the Australian equation for estimating stand volume:

Stand Volume = a + bG+ cH + dG H
	where G is stand BAOB
		H is stand mean height or predominant height
	and a, b, c, d are coefficients.

H.F = V/G.

This relationship has been investigated for various species in a number of countries. Lewis (1954) in New Zealand found that the relationship between form height and stand top height for unthinned P. radiata was almost linear. He used the relationship to form the basis of a VARIABLE DENSITY YIELD TABLE from which present and future volume can be derived (by projecting height and basal area), e.g.

1. Measure stand top height (hdom)
2. Read off the corresponding form height
3. Measure stand basal area G (enumeration or angle count)
4. Stand volume = form height x G.

   V /G 	= a + bhdom
but as 'a' is approximately 0  and 'b' is approximately constant, then :   
  V /G 	= k hdom	
i.e. V  	= k G hdom (the familiar Form Factor Equation)

Application of Stand Volume Tables

• Application should be restricted to the forest population from which the sample stands were drawn.
• Just as one cannot expect a tree volume table to be reliable for estimating the volume of a single tree so, too, the stand volume table is not reliable for estimating the volume of a small stand.

Limitations of Volume Tables

• The basic data are subject to :
  • instrument bias
  • operator bias
  • representativeness of sampling
  • precision of estimate
  • suitability of model
  • representativeness of population
• They do not provide information on recoverable volume or distribution of volume along tree boles (except as indicated by taper tables which give cumulative volume in m3 and diameters under bark at intervals up the stem).

Yield Tables

Yield tables are compiled from relationships between stand variables as dependent variables, and species, age and an expression of the productive capacity of the site as independent variables. Once site index or site quality has been mapped for an area of forest, stand volume at any age, present or future, can be estimated from the yield table without additional field work.

Of course, if actual stand density differs from that listed in the table, a correction must be made to the listed yield table basal area and volume. This adjustment is allowed for in VARIABLE DENSITY YIELD TABLES which include an expression of stand density as a fourth independent variable.

Deriving merchantable volume from total volume estimates

   Merch. vol. ub to 'x' cm dub
   -----------------------
   Total vol. ub

   Stand MVUB to 'x' cm
   ------------------
   Stand TVUB

Stand volume by assortments: Taper tables

• a stand table based on diameter
• a stand height curve, and
• a taper table.

Two main methods of compiling taper tables are in general use (see Chapman and Meyer, 1949. 'Forest Mensuration', McGraw Hill):-

1. Sample trees are sorted into total height classes within dbhob classes and composite stem profiles are made graphically for each class from measurements on the sample trees. Appropriate values of diameter at specific heights are read from these class stem profiles and harmonised over a range of classes. Usually, tree volume tables to various upper diameter limits are compiled from such taper tables and presented with them.
2. A general equation is developed to describe the stem profile and the diameter at any particular point on the stem is derived by substituting appropriate values for the particular tree variables in the equation, one of these variables usually being an expression of tree shape or taper, e.g. form quotient. If the diameters along the stem are each expressed in terms of a standard such as dbh, this leads to an expression of stem form factor for each form class from which the volume of a tree of particular dimensions and form quotient can be derived. This is the basis of the 'form class' approach used by Tor Jonson (Europe), Wright (Canada), Behre (USA) and others (references in Carron, 1968).

The current trend towards production of multiple products from a single tree has created an increased interest in the development of suitable taper equations. A tree taper equation expresses expected diameter along the stem as a function of dbh, total height and height above ground level.

Such equations are particularly useful when used in conjunction with merchantable volume equations. These latter estimate the volume to a specified upper-stem diameter, but utilisation standards often involve log length specifications in addition to a top diameter limit. Veneer bolts, for example, must meet rigid size specifications that reflect the characteristics of the particular lathes used in the manufacturing process. Suppose the specification for veneer bolts requires a length of 2.5 m with a minimum top diameter of 25 cm. How many veneer bolts could, on the average, be cut from standing trees of some specified dbh and total height? Given an appropriate taper equation, this question is easily answered by solving the equation for the diameters at the top of successive 2.5 m bole sections. The number of sections meeting the minimum diameter specification is easily established, and since the end diameters of each qualifying bolt are known, the veneer yield from the tree can be estimated with considerable precision.

No single taper equation can be expected to adequately describe tree form for all species and, in many cases, a single equation will not be adequate to cover all the stand conditions in which a single species may be grown. As a result, many different forms of taper equation have been developed and they have been used for various purposes.

Indirect volume estimation using angle-count sampling

Because of their potential for reducing the cost of forest assessment, both methods need to be tested more extensively under field conditions in a variety of forest types using a taper function to estimate/define the critical height point. Until now, locating this point has been the main factor deterring more extensive testing and use of the methods.

References

• Hooke, R. 1963. Introduction to Scientific Inference. Holden Day Inc., San Francisco. 101 p.
• Demaerschalk, J.P. and Kozak, A. 1974. Suggestions and criteria for more effective regression. Can. J. For. Res. 4(3): 341-348.
• Hummel, F.C. 1955. The volume-basal area line: a study in forest mensuration. Brit. For. Comm. Bull. No. 24.
• Finch, H.D.S. 1957. New ways of using the general tariff tables for conifers. Brit. For. Comm., For. Rec. No. 32, London.
• Clutter, C.L., Forston, J.C., Pienaar, L.V., Brister, G.H., and Bailey, R.L. 1983. Timber Management: A Quantitative Approach. John Wiley & Sons Inc., New York. 333 p.
• Spurr, S.H. 1952. Forest Inventory. The Ronald Press Co., New York. 476 p.
• Bary, G.A.V. and Borough, C.J. 1980. Tree volume tables for Pinus radiata in the Australian Capital Territory.
• CSIRO Division of Forest Research Report No. 11, 1980 (unpublished).
• Lewis, E.R. 1954. Yield of unthinned Pinus radiata in New Zealand. For. Res. Notes Vol. 1, No. 10, For. Res. Institute, NZ For. Serv. 28 p.
• Bitterlich, W. 1984. The Relascope Idea: Relative Measurements in Forestry. Commonw. Agric. Bur., Farnham Royal, U.K. 242 p.
• Kitamura, M. 1962. On an estimate of the volume of trees in a stand by the sum of critical heights (in Japanese). 73 Kai Nichi Rin Ko: 64-67.
• Ueno, Y. 1979. A new method of estimating stand volume (I) (in Japanese). J. Jap. For. Soc. 61: 346-348.
• Ueno, Y. 1980. A new method of estimating stand volume (II) (in Japanese). J. Jap. For. Soc. 62: 315-320.
• Schreuder, H.T., Gregoire, T.G. and Wood, G.B. 1993. Sampling Methods fof Multiresource Forest Inventory. John Wiley & Sons, Inc., NY. 446 p.

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