Stand density and stocking - Oregon State University · FOR 322 Stand density and stocking Stand Stocking: A relative measure (usually expressed as a proportion or a percent) of the

FOR322 Stand density and stocking

• Definitions and concepts• Indices of stand density and stocking

• Number of trees per acre (N)• Basal area per acre (BA)• Total cubic foot volume per acre (TV)• Crown competition factor (CCF)• Curtis’s relative density (RDC)• Reineke’s stand density index (SDI)

• Maximum size-density limit• Size-density trajectory


Stand Density: A quantitative measure of the degree of crowding or competition existing within the stand.


Stand Stocking: A relative measure (usually expressed as a proportion or a percent) of the adequacy of the stand's density for meeting the target management density.

* The target may be expressed as either a minimum or maximum density.

* If a stand's density meets the target, the stand is said to be "fully stocked," if it falls below the target the stand is said to be "under stocked," and if it exceeds the target the stand is said to be "over stocked."


Relative Density: Usually computed by dividing the density of the stand by the "full," "normal" or "maximum" density.

=375

550= 68%


"Stockability": The inherent capability of a piece of ground to carry the "full", "normal" or "maximum" density.

Conditions that can lead to "stockability" problems include:1. Very low rainfall (i.e., rainfall near the lower limit for

the species).2. Limited soil volume due to rock (i.e., the "flower pot"

situation).3. Nutrient deficiency (black spruce bog) or toxicity

(serpentine soil)


Stockability refers to inherent capacity of a piece of

ground.

Caution: Many factors, including past stand disturbances, can cause low density in the current stand

But, in southwestern and eastern Oregon, carrying capacity of maximum stand density has been correlated with plant association (integrated soil and climatic effects).

To a lesser extent, stockability varies even among Westside Douglas-fir sites


Permanent plots in Douglas-fir stands

ln(trees per acre)

ln(q

uad

rait

ic m

ean

dia

met

er)


Competition between trees within a stand has the following effects:

It reduces diameter growth rates

For some species, it reduces height growth rates

It increases crown recession rates

It increases mortality rates


Initial stand density effects on diameter growth:

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120

He

igh

t (f

eet

)

Age (years)

Stand density effect on crown recession

Total tree height

Crown base height, dense stand

Crown base height, sparse stand


Stand density effects on diameter growth:

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120

DB

H (

inch

es)

Age (years)

Wide spacing

Medium spacing

Close spacing


For even-aged (relatively pure) stands, volume is a direct function of:

1) Age2) Inherent site productivity3) Stand density


Stand development can be described in terms of progression of stand density:

Phase 1 : Planting Density Crown Closure

• Trees are growing independently of each other

• Stand growth rate is directly proportional to initial planting density

• No or limited crown lift/crown recession


Phase 2 : Crown Closure Onset of competition induced mortality

• Individual tree growth rates decline• Net stand growth rate continues to

increase substantially• Suppression mortality of branches begins;

crowns begin to rise or recede


Phase 3 : Onset of competition mortality maximum density

• Net stand growth continues at a positive but reduced rate due to loss from mortality

• Mortality induced by continued crown recession; crown size on suppressed trees become insufficient to sustain tree


Phase 4 : At Maximum Density

• Net stand growth rate can continue slowly or may stop based on gross growth rate of the stand and the slope of the maximum size-density line

• Stand growth is a balance between mortality rate and increment on surviving trees


Langsaeter’s hypothesis (note that age is fixed)

Growing stock at a given age






Number of Trees per Acre (N):

EFi = Expansion factor for the ith sample tree= 1.0/PA for fixed area plots= BAF/(0.005454154Di

2) for variable radius plotsPA = Area of the fixed plot in acresBAF = Basal area factor for the variable radius plotDi = DBH of the ith sample tree

n

i

iEFN1


Fixed area plots:

Circular: PA = πR2/43,560R = plot radius in feet

Square: PA = S2/43,560S = plot side length in feet

Rectangular: PA = S1S2/43,560Si = plot side length in feet


Expansion factors:

1/5 ac 5 trees per ac0.1 ac 10 trees per ac1/57.3 ac 57.3 trees per ac0.0175 ac 57.14 trees per ac30-ft radius or0.0648909 acor1/15.406 ac 15.406 trees per ac


N = 15 + 75 + 180 + 155 + 150 + 100 + 1-- + 80 + 55 + 30 + 10 + 15

= 965 trees per acre



• Number of trees per acre (N)

• Basal area per acre (BA)• Total cubic foot volume per acre (TV)• Crown competition factor (CCF)• Curtis’s relative density (RDC)• Reineke’s stand density index (SDI)



Basal Area per Acre (BA):

n

i

ii EFDBA1

2005454154.0

Fundamental side note: Quadratic mean diameter is the diameter of the tree with mean basal area

21

005454154.0

iEF

BAQMD


Handy fact:

21

221

MDS

n

nQMD D

WhereSD

2 = Sample variance of the DBHsMD = Arithmetic mean stand DBH


For the stand represented in Table 1:

And the corresponding QMD:


Note that:

BA = N·(0.005454154·QMD2)



• Number of trees per acre (N)• Basal area per acre (BA)

• Total cubic foot volume per acre (TV)• Crown competition factor (CCF)• Curtis’s relative density (RDC)• Reineke’s stand density index (SDI)



Total cubic foot volume per acre (TV):

n

i

ii EFVTV1

WhereVi = total stem cubic volume of the ith tree




N, BA, and TV as measures of stand density suffer from two problems with respect to interpretation:

1) For a given number of trees per acre, N, further information about average tree size is needed to know how crowded the trees are (e.g., big difference if QMD=5 inches or QMD=20 inches!)

2) For a given BA or TV, further information about number of trees per acre is needed to know how crowded the trees are (e.g., big difference if N=120 or N=1500!)


To address this problem of biological interpretation, other measures of stand density have been derived



• Number of trees per acre (N)• Basal area per acre (BA)• Total cubic foot volume per acre (TV)

• Crown competition factor (CCF)• Curtis’s relative density (RDC)• Reineke’s stand density index (SDI)



Crown Competition Factor (CCF) (Krajicek et al. 1961):

Krajicek et al. (1961) found that a strong relationship existed when the crown width of open grown trees (maximum crown width, MCW) was plotted their DBHs. This relationship has been characterized by the following equation forms:

1. MCWi = b0+b1Di (e.g.,Krajicek et al. 1961)

2. MCWi = b0+b1Di+b2Di2 (e.g., Paine and Hann 1982)

3. MCWi = b0+b1Dib2


0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30 35 40 45

bigleaf maple

red alder

western hemlockDouglas-fir

grand fir

western redcedar

Cro

wn w

idth

(ft

)

Diameter at breast height (in)

Crown width of open-grown trees


Krajicek et al. (1961) then applied this relationship to trees growing in a stand by:

1. Predicting the maximum crown width for each tree in the stand as if it were open grown

2. Computing the maximum crown area assuming the predicted open-grown crown was circular in shape

3. Summing the maximum crown areas for all trees in the stand

4. Dividing this sum by the number of square feet in an acre and multiplying the results by 100 to put it on a percent basis


Actual crown width

Actual and maximum potential crown projection areas:

Maximum potential crown width (MCW)


The implied equation for CCF is:

n

i

ii EFMCWCCF

1

2

560,434

100

n

i

ii EFMCWCCF1

001803.0


Actual crown projection areas:

Crown cover = 75%





CCF = 250%


• CCF computes a % crown closure for the stand assuming all trees are open grown

• Theoretically, a CCF value of 100% indicates that the trees in the stand are just beginning to experience competition (i.e., about 100% canopy cover)

• As CCF increases above 100%, the level of competition between trees increases (but canopy cover remains ≤100%)

• All stands with the same CCF value are experiencing the same level of competition

• CCF is, therefore, a measure of relative density that uses 100% crown closure as the basis for comparisons


Note that if

MCWi = b0+b1Di

Then CCF becomes:

CCF = k1(N) + k2(N)(MD) + k3(BA)

Where,k1 = 0.001803b0

2

k2 = 0.003606b0b1

k3 = 0.330579b12

In other words, CCF can be regarded as a weighted linear combination of N, BA, and MD


One advantage of CCF is that it can accommodate differential effect of species on stand density, because MCW-DBH relationships are unique to each species:

Douglas-fir Pacific silver fir

Same DBH and therefore same BA, but

different contribution

to CCF





• Number of trees per acre (N)• Basal area per acre (BA)• Total cubic foot volume per acre (TV)• Crown competition factor (CCF)

• Curtis’s relative density (RDC)• Reineke’s stand density index (SDI)



Curtis' (l982) Relative Density (RDC):

RDC is derived from:

RD = observed BA / maximum BA

= BA/BAm

Curtis (1982) found that, in stands at maximum density,

the maximum BA (BAm) could be predicted by:

BAm = b1(QMD)b2


Therefore,

RD = BA/[b1(QMD)b2]

Because b1 is just a multiplier, it doesn’t affect relative

density rankings among stands, so Curtis decided to set:

b1 = 1

In coastal Douglas-fir, Curtis found that b2 ranged from

0.45 to 0.50, so he set

b2 = 0.5


The resulting formula for Curtis’s relative density is:

RDC = BA/[(QMD)1/2]

or

= 0.005454154(N)(QMD)3/2

or

= (0.005454154)1/4 (N)1/4 (BA)3/4





• Number of trees per acre (N)• Basal area per acre (BA)• Total cubic foot volume per acre (TV)• Crown competition factor (CCF)• Curtis’s relative density (RDC)

• Reineke’s stand density index (SDI)• Maximum size-density limit• Size-density trajectory


red fir white fir


Reineke`s (1933) Stand Density Index (SDI):

Reineke (1933) found that when the logarithm of N was plotted on the logarithm of QMD for even-aged stands at maximum stand density, the relationship was linear for many species:

LOG(Nm) = b0 + b1LOG(QMD)

Where Nm = Maximum number of trees possible for a stand of given QMD

LOG = Base 10 logarithm


CA mixed conifer

OR/WA Douglas-fir CA

Douglas-fir


LOG(Nm) = b0 + b1LOG(QMD)

Reineke (l933) found the value of b1 to be -1.605 regardless of species.

However, Reineke's (1933) values for b0 do differ substantially between species.


To form SDI, Reineke (1933) assumed that a specific stand falling on a line parallel to the maximum line experienced the same level of competition as all other stands falling on the same line:

LOG(N) = K - 1.605 LOG(QMD)

LOG(QMD)

LO

G(N

)

Size-density limit

(maximum SDI)


Reineke (1933) defined SDI as the trees per acre, N, for a QMD of 10 inches:

LOG(SDI) = K - 1.605[LOG(10.0)]

and solving for K,

K = LOG(SDI) + 1.605[LOG(10.0)]




With fixed slope on relationship between log(QMD) and

log(N), we can determine what SDI is implied by a given

combination of QMD and N:

LOG(SDI) = LOG(N) - 1.605[LOG(10.0/QMD)]

or transforming both side by exponentiating:

605.1605.1

10

0.10

QMDN

QMDNSDI



i.e., the 965 trees per acre with QMD of 7.24 inches are the competitive equivalent of 575 trees per acre with QMD of 10-inches

Density (tpa)

Tre

e s

ize (

Dq

, in

ch

es

)A stand with SDI=595 could have 595 trees with mean diameter of 10 inches

OR10

595

851 trees with mean diameter of 8 inches

8

891



The maximum SDI for a species can be computed by:

LOG(SDIm) = b0 - 1.605LOG(10.0)

Example values of b0 and corresponding maximum SDI by species are:

Species b0 SDIm

Douglas-fir 4.3795 595White fir 4.5241 830Redwood 4.6050 1000


Maximum size-density limits for different tree species and yield classes within species

from Harper (1977) Population Biology of Plants.


Finally, a measure of the stand's relative density (RDR) can be formed by:

RDR = N/Nm

Where,Nm = (10.0b0)(QMD)-1.605

RDR can be more easily computed if the SDI of the stand and the SDIm of the species are known:

RDR = (SDI for the stand)/(Max. SDI for the species)



RDR = 574.7 / 595.0= 0.9659

(Assuming maximum SDI for coastal Douglas-fir is 595)






Reineke`s (1933) maximum number of trees relationship is:

ln(Nm) = b0 + b1ln(QMD)

This relationship can be algebraically manipulated to give:

ln(QMDm) = a0 + a1ln(N)

a0 = -b0/b1

a1 = 1/b1


For Douglas-fir:

a0 = 6.28369

a1 = -0.62305

ln(QMDm) = 6.28369 - 0.62305 ln(N)

This is an example of a maximum size-density relationship that expresses mean tree size as a function of number of trees per acre.


Other measures of mean tree size that have found broad use are mean tree volume and mean tree biomass.

For example, Drew and Flewelling (1979) present the following relationship for Douglas-fir, with slope of -3/2 derived from “-3/2 law of self-thinning”:

ln(MVm) = 12.644 - (3/2)·ln(N)

Where

MVm = Maximum volume per tree in a stand of density N


Rearranging,

ln(Nm) = 8.4293 - 2/3·ln(MV)]

orNm = 4579.446·MV-2/3

Where

MV = mean tree volume in subject stand

= TV/N


Drew and Flewelling (1979) then developed a relative density index (RDI) by:

RDI = N/Nm

= N/4579.446·MV-2/3

= 0.000218367(N)(MV)2/3



RDI = (0.0002183676)(965)(6958.65/965)2/3

= (0.0002183676)(965)(7.211036269)2/3

= (0.0002183676)(965)(3.732488285)= 0.7865

Drew and Flewelling (1979) then constructed a stand density management diagram for Douglas-fir based on their maximum size-density line and relative density index


Drew and Flewelling stand density management diagram

(Note isopleths for mean stand diameter and height)


Drew and Flewelling proposed the following hypotheses concerning stand growth:

1) At densities below crown closure (less than 0.15 relative density) growth per unit area is proportional to density.

2) At relative densities beteen 0.15 and 0.40, growth per unit area increases with density, but growth per tree declines.

3) At relative densities between 0.40 and 0.55, growth per unit area is unaffected by density.

4) For relative densities greater than 0.55, gross growth is the same as in the 0.40 to 0.55 region, but net growth may be considerably less than this if substantial mortality has occurred.


Drew and Flewelling stand density management diagram

Density

Tre

e s

ize

Zone of competition

Mortality (relative density=0.55)

Minimum recommended

density to maintain site

occupancy (relative

density=0.35)

Maximum SDI (relative

density=1.00)

Approximate crown

closure (relative

density=0.15)



Similar stand density management diagram, but based on QMD rather than MV (like Reineke but with axes switched)

600

from Long et al. (1988)




• Maximum size-density limit

• Size-density trajectory


The maximum size-density line only defines the maximum size that trees can attain for a given N.

Implied in the concept of a maximum size-density limit is the development of stands in a way that they approach the maximum size density limit and track (or become asymptotic to) this limit as they self-thin and suppression mortality proceeds.

The path that stands follow has been referred to as

the size-density trajectory.


Time trajectory of stands over stand density management diagram (from Drew and Flewelling 1979)


Red alder


Smith and Hann (1984) modeled stand trajectories as they approached the maximum size-density limit as:

ln(Y) = a0 + a1ln(N) – (a0a2)exp{-a3[ln(N0) – ln(N)]a4}

WhereY = Mean tree sizeN = Current number of trees per acreN0 = Number of trees per acre at initiation of

mortality)a0, a1 = Parameters defining the maximum size-

density line


Important points about the size-density trajectory:

• The trajectories describe unthinned, self-thinning stands; thinned stands behave differently.

• The specific trajectory a stand follows depends only upon the initial trees per acre, N0

• The shape of the trajectory is not a function of productivity; productivity affects how fast the stand moves along the trajectory, but not the path

• The size-density trajectory is static – it does not predict the rate of movement of a stand along the trajectory; a dynamic model requires addition of a model for rate of mortality.

Density

Tre

e s

ize

Heavy thinning

Light thinning



Can also model the size-density trajectory with QMD instead of mean volume or biomass.

One approach is to use the reciprocal of the Reineke slope : 1/-1.605 = -0.62305

Under this slope, stands growing along their maximum SDI will increase in basal area over time.

With a slope of -0.5, stands growing along their maximum SDI will have constant basal area.


Why are the maximum size-density limit and maximum size-density trajectories important?


Why are the maximum size-density limit and maximum size-density trajectories important?

• Stand density management• Current assessment of stand density relative

to management objective• Design of thinning regime to meet stand

management objective• Constraint on growth models; check on

individual-tree mortality model

FOR322

End 30 January 2015

Stand density and stocking

Documents

Stand density and stocking - Oregon State University · FOR 322 Stand density and stocking Stand Stocking: A relative measure (usually expressed as a proportion or a percent) of the