_ Agriculture United Department Statesof Stockability, Growth,

_o_t and Yield of the

North Central

Forest Station Experiment Circumboreal Aspens

Research NC-321 Paper (Populus tremuloides

_ Michx., P. tremula L.)

D, A. Perala, R. A. Leary, and C. J. Cieszewski
The policy of the United States Department of Agriculture Forest Service prohibits

discrimination on the basis of race, color, national origin, age. religion, sex, or

disabflRy, familial status, or political afflliaUon. Persons believing they have been

discriminated against ha any Forest Service related activity should write to: Chief,

Forest Service, USDA. P.O. Box 96090, Washington. DC 20090-6090.

North Central Forest Experiment Station

Forest Service--U.S. Department of Agriculture

1992 Folwell Avenue

St. Paul, Minnesota 55108

Manuscript approved for publication January 10, 1995

1995
Stockability, Growth, and Yield of the Circumboreal

Aspens (Populus tremuloides Michx., P. tremula L.)

D.A. Perala, R.A. Leary, and C.J. Cieszewski

We show elsewhere (Perala et al., in review) that If we let W be proportional to D^2*H (Clutter et

quaking aspen (Populus tremuloides Michx.) and a/. 1983), where

its Eurasian counterpart, P. tremula L., form a

single circumpolar superspecies when viewed H = mean stand height,

from the standpoint of self-thinning rates and D = quadratic mean d.b.h.,

stockability. Here we expand our examination to

the d.b.h.-age relationships and to growth series and let H be proportional to D^0.8 (Curtis 1970),

measurements from permanent plots of aspen [1] becomes the limiting relationship,

stands of varying densities reported in the literature. We also attempt to account for the curvilin- [2] D = k1*Nn-0.54.

ear trend in the self-thinning relationship we

detected in young stands that forced us in our That is,

first analysis to truncate our usable data set to

older stands. The resulting equations satisfy the [3] W::DA2*H::DA2.8::N^-I.5

need for a framework to study variation in aspen

stockability. The equations can give useful becomes

regional estimates as well, but will need refitting

to local data to satisfy needs for finer resolution. [4] D = k1*N^(-l.5/2.8)

= kl*N^-0.54

SPACING INDICES REVIEWED

after introducing a rate parameter. The reciproThe several spacing indices offered in the litera- cal relationship of [4],

ture are essentially the same (Curtis 1970,

Bredenkamp and Burkhart 1990). The index [5] N = (D/kt)^(1/-.54)

receiving the most attention recently, the = (D/k1) ^- 1.85

self-thinning rule, is simply = (I/kt) ^- 1.85"D ^- 1.85

= k2*D ^- 1.85

[I] W = ko*N ^- 1.5

is slightly more nonlinear (power of- 1.85 vs.

where -1.6) than noted by Reineke (1933).

W = mean weight The lq-parameters in these relationships were

N = stand density thought to be species-dependent, but not site- or

k o = rate parameter age-dependent (Reineke 1933, Yoda et al. 1963,

Drew and Flewelling 1977). Weller (1990) and

Lonsdale (1990), however, offer evidence that the

D.A. Persia, Research Forester (retired), North growing environment, as well as plant genetics,

Central Forest Experiment Station, Grand Rap- can alter both the rate and power parameters.

ids, Minnesota. The power, -3/2, should be considered a generality that is seldom achieved.

R.A, Leafy, Principal Mensurationist, North

Central Forest Experiment Station, St. Paul, Variation among stands of a species in biomass

Minnesota. packing has been termed "carrying capacity"

(Strub and Bredenkamp 1985, Harrison and

C.J. Cieszewskl, Mensurationist, Northern Daniels 1988), "average mass density" (Verwijst

Forestry Centre, Edmonton, Alberta.
1989), and "stockability" (Debell et al. 1989). We Sugita 1989). Indeed, the relationship has been

will use the latter term. found useful in modeling productivity of several

forest types, including P/nus radiata D. Don

A stand might be considered a packing space for (Drew and Flewelling 1977), P. contorta Dougl.

a limited mass of water-conductlng xylem (deter- (Flewelling and Drew 1985), P. resinosa (Smith

mined by an upper leaf area limit). The dead 1986, Smith and Brand 1988), Pseudotsuga

non-conducting heartwood provides only me- menziesii (Mirb.) Franco (Drew and Flewelling

chanical support but determines, in turn, how 1979, O'Hara and Oliver 1988), Alnus rubra

much crown can be supported by the stem (Dean Bong. (Smith 1986, Hibbs 1987), Thuja plicata

and Long 1986). Therefore, the size-density Donn. (Smith 1989), and Eucalyptus grandis

limiting (self-thinning) relationship can be ap- (Hill) Maiden (Bredenkamp and Burkhart 1990).

plied as mean d.b.h., height, weight, or volume

dependency on stand density. But not all these METIIOI_

relaUonships are independent of site and age.

For example, consider two different sites with McFadden and Oliver (1988) integrated three

stands of the same age, mean d.b.h., and stock- two-dimensional stand models---tree size/tree

lng density. The stand growing on the better site number, tree size/stand age, and tree number/

has taller trees and therefore greater mass than stand age--into a single compatible three-dtmenthe other stand. On a traditional Relneke dla- slonal stand growth conceptual model. Selfgram, both stands would be represented by a thinning stands in their model might follow any

single point: but on Czarnowskt's and Yoda's one of three tree size/tree number dynamic

diagrams, the stands would be represented by trajectories, depending on initial stand density

two points depicting different competition levels and age. In trajectory I, all stands follow the

and likely different survival (Czarnowskl 1961, same trajectory at the same age; in trajectory II,

Yoda et al. 1963). The better site with taller trees stands follow the same trajectory but at different

wouldbe more tightly packed. Furthermore, ages; and in trajectory rn, stands of different

given two more stands having the same height, ages or with different initial tree numbers follow

d.b.h., and stocking density, but different ages different trajectories. We Infer that site quality

{yet old enough to have attained maximum leaf adds another dimension to stand growth (by

area), the mass of these two stands would likely affecting the rate of stand change) and to maxidiffer because tree taper usually decreases with mum density as well depending on stand history.

age. These stands would be represented by

single points on Relneke's and Czarnowskl's Weller (1990) distinguished two discrete apdiagrams, but again by two points on Yoda's. proaches to modeling based on definition of the

The site with older trees would be packed more self-thinning line: (a) the dynamic thinning

efficiently. The density relation we use deter- line (the log-linear self-thinning trajectory

mines whether the stands appear different in tracked by individual crowded stands), and (b)

compeUtive status and mortality rates. Because the species boundmsr line (the upper boundary

the Yoda relationship seems the most complete, self-thinning line attained only in extreme cases).

It is reasonable to conclude that the Reineke We developed generic equations for aspen growth

relaUonshlp can benefit by extension of the and yield based on Weller's species boundary line

factors, site and age. concept applied to the McFadden and Oliver

model, trajectory III.

Self-thinning relaUonships have their critics

(Weller 1987; Zelde 1987, 1991: Lonsdale 1990), Data Sources

and have stirred lively debate (Osawa and Sugita

1989; Weller 1990, 1991): nevertheless, no useful Stand data

alternative has been offered. Indeed, Zeide

(1991) states "the self-thinning law is perfectly Data for modeling stand d.b.h, growth, tree

correct and even trivial within the assumptions survival, and yield of quaking aspen, Eurasian

explicitly stated by its authors v/z. constant aspen, and their hybrids were extracted from the

crown form and horizontal canopy closure, published literature along with some data previously unpublished or limited in distribution

Self-thinning relaUonships perform best when (table I). Some data for largetooth aspen (P.

applied strictly to fully occupied sites {Osawa and

2
Table I .--Data sources by species and provenance

Number of cases Variable and ran_ge

Author Yield Dynamic Tabular Site index Age Mean d.b.h.

m @50 y yr cm

P. tremuloides

Alaska

Gregory and Haack 1965 37 11-23 20-100 2-21

Prairie Provinces

Bella 1975 9 20 3-6 1-2

Beila and DeFranceschi 1980 33 16-24 6-40 1-11

Kirby et aL 1957 27 13-19 20-100 4-29

Pike 1953 4 3 17-18 35-55 8-16

Steneker 1969 10 21 19-21 14-30 4-15

Steneker 1974 29 51 16-21 11-44 3-23

Rocky Mountains

Baker 1925 5 17 100-140 24-33

Bartos and Lester 1984 3 8-10 54-82 13-21

Crouch1981 1 17 66 20

Jones and Trujiilo 1975 6 12-20 22 4-5

Kemperman and Barnes 1976 2 9-15 100-105 23-28

Schier 1975 5 9-12 70-92 15-23

Schier and Smith 1979 1 18 67 21

Waiters et al. 1982 3 13-19 60-70 21-28

Great Lakes

Barnes 1969 3 12-19 16-36 4-13

Brownand Gevorkiantz 1934 27 12-24 20-80 3-36

Day1958 7 3 18-24 10-25 3-11

Graham et aL 1963 16 14-26 25-50 8-30

Hubbard1972 4 6 27 7-24 3-14

Noreen1968 9 5 24 4-20 1-10

Perala1974 7 21 2-10 1-4

Perala1978 7 48 23-25 13-53 4-31

Perala1979 2 23-24 4-8 1-3

Perala1984 2 23-27 12 5

Peraladatafiled 6 19 24 15-39 5-17

PeralaandAlban1982 2 21-22 40-49 18-24

PeralaandLaidly1989 24 25-31 5-21 3-13

Plonski1960 51 16-24 20-100 6-33

Schlaegel1971,1972 60 18-24 10-62 5-35

Schlaegel 1975 1 24 40 18

Schlaegel and Ringold 1971 8 26 37-47 17-24

Newfoundland

Page 1972 8 17 20-90 5-22

P. tremula and hybrid

Scandinavia

EIfving 1986 4 34 9-32 7-25

Haugberg 1958 34 69

Vuokila1977 2 11 22-23 11-48 3-25

Russia

Vincent et al. 1950 50 11-24 10-100 1-34

TOTAL P. tremuloides, tremula 152 339 254 8-34 2-168 1-43

P. grandidentata

Michigan

Graham et aL 1963 19 14-26 25-60 9-27

Koerper and Richardson 1980 3 16-27 52-60 14-19

3
grandlclentata Michx.) were used to validate the If site index was missing or given in a base year

models. These data comprised three sets: other than 50, it was estimated from accompanying or local tabular or graphical values. If these

(a) Yield set--singular empirical observa- were unavailable, site index was estimated from

tions of aspen yield reported for

unmanaged fully stocked stands. These [7] S = Hd/{1.48*[1-exp(-0.0214*A)]^0.9377}

were usually controls or initial conditions

prior to experimental treatment. This set (Gevorkiantz 1956, Lundgren and Dolid 1970)

was used to model the relationship of where

variables acting upon stocking and the

self-thinning relationship. The resulting S = site index, m at 50 yr

equation defined the terms and their H d = mean total height of dominants and

parameter values needed to model the codominants, m

dynamic set. A = total stand age, yr

(b) Dynamic set--singular or serial em- Other data, such as sapwood area or leaf area,

pirical observations of aspen growth for were much too sparse to be useful. 

both experimentally treated and untreated stands. Thinning was the only These data conformed generally to the d.b.h.-age

treatment specified except that Perala and and the self-thinning relationships (figs. 1,2).

Laidly (1989) fertilized as well, and the

aspen hybrids were planted (Elfving Env/_nment data

1986). This set was used to model stand

growth and survival. Because the data sets are so wide ranging, we

expected that latitude, longitude, elevation, and

(c) Tabular set--smoothed yield data other environmental variables might account for

based on many observations over a some regional variation. The edaphic environgeographical area (e.g., Brown and ment was seldom described adequately, but

Gevorkiantz 1934, Plonski 1960). These climatic data were sometimes given or could be

are found in "normal" or "average" yield inferred from other sources, vtz.

tables. This set was used to confirm and

validate the model determined from the total annual solar radiation, MJ/m^2*y

yield set. These tables were constructed (Budyko 1982)

in different ways (hand-fitted or statistically derived regressions) from data taken mean July air temperature, C (Hambidge

from subjectively chosen field plots. 1941, Anon. 1984)

Largetooth aspen yield and tabular data

are included in this set. total annual precipitation, mm (Hambidge

1941, Anon. 1986)

Minimum stand data needed were tree number

(all trees taller than 1.37 m), quadratic mean annual runoff, mm (Geraghty et cd. 1973,

d.b.h., total age (but not younger than attain- Pearse eta/. 1985)

ment of maximum stand density, usually 2

years), and site index. If either tree number or These are based on long-term averages, often 30

d.b.h, was missing, it was derived from given years. For tabular data, these values were

basal area according to estimated for the centroid of aspen distribution

within the stated provenance.

[6] D = (12732*B/N)^0.5

Model Development

where

Standard linear regression procedures (Weisberg

D = quadratic mean d.b.h., cm 1985) were followed in preliminary modeling to

B = stand basal area, m2*ha -I

N = stand density, stems*ha -1
O. ,m, O. •

o°_eDo . *q'.

lO- . : - lO- • " • -

LoJ_ ..._:, [b1_ -_,.,'"..

8 - " - 8 - * -

• _ :_._.,._:.

• ¢_51

oO •

[ I 6 - * - 6 - -

t-!

rj_

_, 4 I ! I I 4 i i i I , • i

IT] -1 0 1 2 3 4 --1 0 1 2 3 4 • !

12 n / u u ...... 12 u t n i

OII •

W_

_ I 10- - 10- "":. -

8- -- - B-

• qlIL__di4_.-'.I"iilb

[c] "-•" [d] ...._.

6- ._ 6 - .

4 I I I I 4 I I I I

--1 0 1 2 3 4 --1 0 1 2 3 4

in DBH

Figure 1.--The relation of stand density to quadratic mean d.b.h. (double-logarithmic scale)for (a) the

entire data set, (b) the yield set, (c) the dynamic set, and (d) the tabular set.

identify significant variables. To linearize rela- plots. Variables seemingly related to the deviationships and to stab!!Ize variances, we trans- tions were introduced to the model, and new

formed models to the logarithmic mode. Devia- equations were solved. Variables were retained

t.ions about these models were routinely plotted only if p<0.01. Nonlinear procedures (Wilkinson

over environmental variables, and screened for 1988) were used to develop the final equations

treatment, species, and lack-of-fit trends in that are expressed along with their statistics and

scatter diagrams, box plots, and probability interpretation in the original arithmetic mode.
4 u n u u i 4 1 / i i u

-

2 - ".'_',- t__ _ 2 - _'_" [b] -

__tSri ; -LaJ

• :." .-'.t

1 - ".•_. o . - 1 - "-- | -

OQI_ •

• | • e

0 - I'• - O- ! -

v • • •

r-"l -1 i i n I i -1 i,, I ! t I i i

0 1 2 3 4 5 6 0 1 2 3 4 5 6

4 I i , , i 4 , , i i ,

,, l

• • ,. I_.o ee

"" .t.l_l..

"" '!'1":'"

• i O • II o

2 - 2 tle* * -

. : I-."

ibOV • •

1 - " - 1 - I " -

[c] I "• t " [d]

I

0 - - 0 - " -

--1 q I ! I I --1 I I ! ! I

0 1 2 ,3 4 5 6 0 1 2 3 4 .5 6

in AGE

Figure 2.--The relation Of mean stand d.b.tr to stand age (double-logarithmic scale)for (a) the entire

data set, (b) the yield set, (c) the dynamic set, and (d} the tabular set. °

Yield Long 1992). Crown length in self-thinning

stands is limited by tree height until a light

The self-thinning relationship tailed off for stands extinction threshold is reached where lower

smaller than just a few cm d.b.h. (fig. 1) as.we branches begin to die (Beekhuis 1965). Thereafnoticed earlier (Perala et at, in review). Aspen ter, crown length, foliar area, and, by implication,

quickly attains horizontal canopy closure and k 2 should remain at some asymptotic maximum

can be self-thinning in the second year (Perala value. Moreover, crown shape changes with age

1984). Early leaf area deficiency seems to be (Zeide 1991). For example, Verwijst (1989) found

caused by immature canopy structure (Dean and the self-thinning power, -1.37, for Betula
pubescens Ehrh. deviated significantly from -1.5 g = growth power parameter

because crown depth diminished as trees grew ei = deviation from estimate, geometric

taller. To maintain constant leaf area during mean= 1

self-thinning, the crowns expanded radially at

the expense of packing the space occupied. Similar to [8], the growth relationship [10] has a

Crown length data were unavailable to us, but constant power (g), but the rate paramater

the effect can be estimated from tree height, increases with site index and with mean July

which is proportional to DA0.8 (Curtis 1970). Of temperature.

course, tree height is also related to stand age,

but the high collinearity between d.b.h, and age Exploratory fitting of the tabular data set con-

(fig. 2) would lead to ambiguous hypothesis firmed models [8] and [I 0] were appropriate

testing of associated Student t-values, choices; no better models were discovered.

In addition to mean d.b.h., stocking was related _am_: ful/y stocked stands

to climate variables, but better related to mean

July air temperature than to precipitation as we The self-thinnlng relationship is static and

found in the truncated data set (Perala et at, in therefore is useful only to describe the standing

review). The best-fit nonlinear model was crop of fully stocked stands. To function as a

survival model, a dynamic element is needed.

[8] N = a*(l-cAD)*dAJ*DA-b*el Tait (1988) and Tait et al. (1988) used the relationship in dynamic form with good results for

where fully stocked P. menziesii and P. contorta. Model

[8] provides the basis to develop such a model for

N,D = stand variables already described fully stocked stands. If we linearize [8] by transJ = mean July temperature, 17 C maxi- forming to the logarithmic mode, and differentimum ate instantaneous logarithmic relative survival,

a,c,d = estimated parameters, components

of k2 d(InN),/.e. In(N2/N l)

-b = self-thinning parameter

el = error about estimate, geometric with respect to log relative d.b.h, growth (ignoring

mean=I (I-cAD)for the moment),

That is, the self-thinning relationship [8] has a d(InD), Le. In(D2/D I)

constant power (-b), but the rate parameter (k 2)

increases as crown length increases (a function we obtain the partial derivative

of d.b.h.) and diminishes with temperature.

[I i] In(N2/NI)/In(D2/Dl)=-b

The fundamental component of model [8],

where

[9] N= a*DA-b

InN = log number of trees per hectare

expresses the self-thinning relationship and present at log d.b.h. D

trajectories I and II of McFadden and Oliver subscript2= future value

(1988). Introducing the asymptotic effect of subscript1= starting value

crown length, (1-cAD), and temperature raises -b= self-thinning power

the model to trajectory Ill.

We consider that the self-thinning power has two

Mean d.b.h, was related to age, site index, and components,

July temperature,

b = b'+w

[10] D= f*AAg*j^J*qAS*ei

where

where

b' = competition component

DJk,J,S= variables already described w = pathogenic component

f,J,q=estimatedratecomponents 7
Dynamics: understocked stands

In reality, other components contribute to the

self-thinning power, including b'*w and episodic The self-thinning relationship needs an asympevents, but these axe fax beyond the scope of this totic approach of understocked stands to the

analysis, maximum size/density constraint (Hara 1984,

Smith and Hann 1984, Lloyd and Harms 1986,

The reciprocal differential, log relative d.b.h. Harrison and Daniels 1988, Smith and Brand

growth with respect to log relative survival, 1988). Stocking is defined as "a more or less

subjective indication of the number of trees as

[12] In(D2/D1)/In(N2/N1)=I/-b compared to the desirable number for best

results" (Ford-Robertson 1971). Here we express

is also legitimate. Although simultaneous solu- relative stocking (RS) using given stand density

tion of [11] and [12] is possible, dual application compared to the species boundary line (Clutter et

to a growth prediction system is difficult. Rather, aL 1983), expressed as a decimal.

we reasoned that because mortality in selfthinning stands is growth-driven, and that Models [14] and [15] are deficient for under- 

growth is time-driven, instantaneous log relative stocked stands, and the power parameters must

d.b.h, growth, be adjusted accordingly. Consider the conceptual, simplistic growth and survival trajectory of

d(InD), i.e. In(D2/D 1) a sparse stand depicted logarithmically in figure

3. At age A 1, RS is much less than 1.0, competishould be specified with respect to log relative tion for site resources is lacking, radial growth is

elapsed time, maximum at g+p, where

d(InA), i.e. In(A2/A l) p = residual growth potential,

to give and only background mortality (w) is at work. At

A2, RS is still sufficiently <1.0 that -b=-w; how-

[13] In(D2/Dl)/In(A2/A1)=g ever, radial growth will have slowed because it

where 9

L_

InD=log quadratic mean d.b.h, at log age A o

g=growth power v

T 8 inning line Integrating [11] back to the nonlinear mode, L. @

including the previously ignored (1-c^D), gives

the proportional equation for survival, m

[14] N2fN1*(D2/Dl)^-b*(1-c^D2)/(l-c^D1)*el }- %- 7 A1 A2A3_"'_

o Understocked A4"_

based on the self-thinning relationship. Like- _ stand "_

wise, integrating [I 3] gives the proportional zE3 AsN_

equation for radial growth, _ 6

Z A6

[15] D2=D1*(_/Al)^g*el

based on the d.b.h.-age relationship. 5 J

1.5 2.5 3.5

The observed future value is the starting value

for each successive iteration in our analysis. In In Dbh, cm

[14], ffD2=D l, then N2=N1; in [15], ffA2=A1,then

D2=D I. These are desired attributes of dynamic Figure 3.mConcepb__a! survival trajectory for an

stand equations (Clutter eta/. 1983). understocked stand inferred from flg. 1 (c).

8
Figure 5

has surpassed the threshold of resource limit.

From A 3 to A4, -b asymptotically departs from -w 1.2

and increases exponentially while radial growth 0

slows further. From A4 to A5, mortality asymp- o o©

totically approaches that given by the self-thin- © ©

ning line (-(b'+w)) as RS approaches 1.0. At © ©

RS= 1.0, from As to A_ and beyond, stand dynam- O1.1 o o

ics adhere to self-thinning and d.b.h.-age trajec- _ r_co

tories. Thus, from sparse to dense stands, the ...f OO O 0

doubly asymptotic values of the self-thinning = __ Oo_ °

power would range from 0 to -b, and values of 9 _o_ o

the growth power would range from g+p to g. _ o

_0 0 0 _:D

.8 1.o o ¢bo__- o: • moo

Clearly, some factor (call it F)must be introduced = oo___:_ _ o

to allow this flexibility: = o

°° _ "--o

[16] N2= NI*(D21DI)^(-w-b'*Fb)*(1-c^D2)I OOo o"k_ o

(1-cAD1)*eland o

0.9 i

[17] D2- Dl.(A2/A1)^(g+p.F_.el 0 I 2

Relative stocking

where Figure 5.--Residun!s afler fltting dynamic data

0<=Fb<= 1, 0<=Fp<= 1 set to model [15] in relation to relative stockO_

estimated from model [I 9]. The fitted trend is

From figure 4, and with D 2 given, we infer that Fb the modified Weibullfunction (eq. [21]) solved

depends on potential RS, in eq. [17].

Figure 4

[18] RS2= N1/[a*(1-c^D2)*d^J*D2^-b]

1.1

10- __ _o Because must [19] beRSestimated ]D=2 isN1the/[a*(l-c^D dependent from 1initial )*d^J*Dvariable RSl^-b] , in [I 7], Fp

_ o Both doubly asymptotic relationships of F to RS

0.9- OCO cc_ o Weibull (Figs. 4 and function 5) can(Fisher be easilyand mTippett odeled 1928 with,the

°°--_>"_o o Weibull 1939). A common form based on Yang et

Q 0.8 - o O _ci_ a/. (1978), O

© .0 0

O _ [20] Fb= l-expl-l(RS2/r}^sl}

_ o o a_e_,oo_ oe

_. 0.7 o where

t_® O O

o

o oo r = rate parameter

06 _ J s = shapeparameter

0 1 2 3

Relativestocking is proper for [16]. A modified form,

Figure 4.--Resiclua!s after fitting dynamic data

set to model [14] in relation to relative stocking [21] Fp= exp{-[(RS1/r)^s]}

estimated from model [18]. The fitted trend is can model [17]. The Weibull function, noted for

the Weibull function (eq. [20]) solved in eq. [16].

its rich expression of curve shapes, is especially

9
adept at fitting the doubly asymptotic nature of RF_UI,TB AND DISCUSSION

biological processes as well as probability distributions for which it was first developed (Yang et Stocking and Yield

oZ 1978).

Aspen yield is a self-thinning relationship in

Eq. [16] tests whether RS alters trajectory Ill of general (fig. l(b), (d)) and in particular according

McFadden and Oliver (1988). Eq. [17] tests the to trajectory Ill (tables 2 and 3, eq. [8]). The b

effect of RS on radial growth, parameter, -1.66, lies closer to the parameter

value -1.56 for Alnus rubra, a similar lightAnalysis demanding pioneer species, than it does to the

- 1.92 for Pseudotsuga rnenziesii (Puettmann et al.

The yield and dynamic data sets were merged, 1993). It is significantly (p<O.05) smaller than

and a two-step analysis provided compatible both the theoretical -1.85 (eq. [5]) and the -1.96

yield and dynamic equations that have good fit we found with the much smaller data subset

staUstics with minimal bias among data sets. (Perala et aL, in review), but not significantly

First, logarithmic transformations of stocking larger (p>0.10) than Reineke's -1.6.

models [8] and [16] were fit simultaneously (by

nonlinear regression weighted by the inverse of Why the rather large discrepancy between the

whole-set variance, Wilkinson 1988) to give an b-values of this and our other analysis? Several

average solution of parameters in common, factors can contribute. First, another term

Dummy variables (0,1) were used to control (1-c^D) was introduced to the model. However,

model assignment to data set (Weisberg 1985). the self-thinning relationship seems little conThen d.b.h, models [10] and [17] were fit as founded with this term that estimates 96 percent

above except that only the yield set needed of leaf area is acquired by 9.0 cm d.b.h., the

logarithmic transformation, and RS l (Eq. [19]) lower limit used in our other analysis (Perala et

was calculated using parameter values solved for aL, in review). Thus, the bulk (about three-

[8] and [16]. fourths) of the data is little influenced by the

term. Furthermore, our predictions of leaf area

We also ran weighted nonlinear analyses without attalnment take on some realism when compared

logarithmically transforming these models, with to two lines of evidence from the literature.

little change in outcome. Apparently, logarithmic Ruark and Bockheim (1987) studied aspen leaf

models adequately stabilized the distribution of area over a chronosequence of 8 to 63 years on a

e_. single soft series. Their leaf areas agreed closely

with our estimates from about 6 cm d.b.h.,

Verification and Validation stabilizing at about 14 cm:

Model behavior was evaluated by studying trends Leaf area attained

and patterns in bias and accuracy (Burk 1986), Age D.b.h, Ruark and Bockheim This study

[22] bias percent={ lO0*[geomean(Yp/Y)- 1]} (yr) (cm) (percent) (percent)

8 3.6 59 71

[23] accuracy percent={ 100*[geomean(Yp/ 14 5.5 70 85

Y)-I]}; (Yp/Y) inverse when <1 18 6.1 85 88

22 9.0 not studied 96

where 32 14.1 1O0 99

63 19.2 100 100

Yp = predicted arithmetic value

Y = observed arithmetic value

Another study by Verry (I 987) bears indirectly on

The yield and tabular data sets were used to this question. Streamflow was greatly enhanced

evaluate patterns of fit for equations [8] and [10] during the first several years after a mature

by study and provenance summary. The dy- aspen-dominated watershed was clearcut. Flow

namic set was used similarly for [16] and [17].

Only the results from the tabular set constitute

independent validation.

10
Table 2.-- Equation parameter values and statistics a

Parameterb Equation Value t-value Probability

a [8] 4.049x10 s 25.21 0.0000

w [8],[16] 2.811x10 -1 4.61 0.0000

b' " 1.376 35.48 0.0000

c " 7.022xl 0-1 6.72 0.0000

d " 8.213x10 -1 6.15 0.0000

r [16] 1.215 24.47 0.0000

s " 4.950 7.58 0.0000

f [10] 4.807x10 _ 10.51 0.0000

j " 1.071 3.88 0.0001

q " 1.036 7.88 0.0000

g [10],[17] 1.009 82.62 0.0000

p [17] 2.509xl 0-1 7.54 0.0000

r " 3.806xl 0-1 21.58 0.0000

s " 4.751 3.95 0.0000

==Calculatedfrom serial correlationcoefficientof residuals(0.315 for[16], 0.104 for[17]) accordingto Stewart-Oatenand

Murdoch (1986).

bparametersa andf adjustedforlogbias accordingto Baskerville(1972).

Table 3.--Eqtmt/onfit stat/st/cs a

Equation

Statistic [8] [10] [16] [17]

Syb 2259 1.94 81.4 0.493

percentb 26.3 19.6 5.49 3.07

adj. R2 0.961 0.905 0.994 0.995

n 152 154 328 333

"Determinedby data setfrom residualsgeneratedbythe combinedanalyses.

bat meanY, calculatedaccordingto Baskerville(1972).

11
subsided to near, but not at, preharvest levels by power parameter (table 2) not significantly

15 years as the aspen stand developed into the greater than 1.000 (p>0.40), meaning virtual

sapling stage. The retum of flow toward the linearity.

preharvest level was related to greater evapotranspiration and rainfall interception by in- Age accounted for 85 percent of the total variabilcreasing leaf area. Increased water storage in ity in diameter, site index another 4 percent, and

decomposing logging debris also aided the recov- mean July temperature another 1 percent. The

ery so that inferred leaf area is exaggerated. This parameters for these variables indicate for each

and the Ruark and Bockheim study provide doubling of age, diameter doubles; for each

estimates that bracket our predictions of leaf degree warmer up to 17 C, diameter increases by

area attainment. A more robust validation 7 percent; and for each 3 m improvement in site

awaits addiUonal measurements relating to leaf index, diameter increases by 11 percent. It is

area. Specifically, does aspen leaf area culminate well known that as site index increases, d.b.h.

when d.b.h, averages 15 cm, as [8] predicts? invariably increases, other factors being equal

This is equivalent to the 17- to 18-m mean stand (see any yield table).

height attained in 30 to 40 years on the better

sites (Brown and Gevorkiantz 1934, Gregory and Stand Dynamics

Haack 1965).

The survival and growth models ([16] and [17],

A second factor is the dominance of the data by tables 2 and 3) fit well the wide range of data

the growth set that introduces another term, the sources. The S-shaped Weibull function (eq.

Weibull function. Fitting the function unavoid- [20]) was especially critical to fitting the survival

ably affects the value of all other parameters model, improving Sy.x by 75 percent over eq. [14].

through the least squares process, although the

great plasticity of the function suggests such The modified function (eq. [2 I]) improved Sy.x for

influence to be minimal, radial growth by a more modest 15 percent over

eq. [15]. This function acting on the parameter p

Finally, the nature of the yield data may be produces the near-linear relation of mean stand

inconsistent with the species boundary line. diameter over age noticed by Schlaegel (1971)

Investigators selecting "normal" or "fully stocked" and Bella (1972) when F=0 in self-thinning

stands routinely bias their choices by discarding stands. In contrast, the value of p (table 2)

stands with "excessive" mortality; /.e., those indicates strong nonlinearity (g+p= 1.26) in

growing at or near their dynamic thinning lines open-grown stands when F= 1.

(Weller 1990). Stands varying in the slope and

intercept of the dynamic thinning line and in The rate and shape parameters indicate the

location relative to the line, and investigator bias WeibuU function is sharply asymptotic for both

proportional to size could easily produce trend growth and survival. These functions confirm

data disproportional to the species boundary that both radial growth and survival were less in

line. If correct, our analyses show that the dense stands. Neither site nor July temperature

species boundary line has a lesser power than is is a growth factor here (except that temperature

suggested from the trend of normal data. helps determine relative stocking) because each

is already integrated into diameter at a given age

Despite all this, the utility of the self-thinning (e.g., Buckman 1962).

relationship is clear. D^-b alone in [8] accounts

for 93 percent of the variation in stand density Climatic Relations

compared to the 96 percent accounted overall

(table 3). The fit of all equations was improved significantly

by incorporating the climatic variable, mean July

Mean d.b.h, of fully stocked stands plotted over air temperature, to compute stocking (table 2).

age produces a strong and apparently linear The coefficient (d) indicates that for each 1 C

relationship (O'Connor 1935, Schlaegel 1971, increase in temperature up to 17 C, stockability

Bella 1972). We found the relationship [10] diminishes by 18 percent. Aspen longevity is

linear on logarithmic scales as well (fig. 2), with a well known to depend in part on a cool climate

12
(Shields and Bockheim 1981, DeByle and more sensitive to stress than is survival. Again,

Winokur 1985, Perala 1991). Water deficits the stress is probably more related to water

associated with warmer temperatures deficiency than to temperature per se (Jones and

(Thornthwaite and Mather 1955) seem to be the Schier 1985).

limiting factor (Fralish 1972), probably by limiting leaf area (e.g., Gholz et al. 1976, Grier and Validation

Running 1977). A favorable water balance allows

more foliar transpiring surface that must be Pattem analysis revealed that the dynamic

supported by greater sapwood area, therefore models performed well with little bias and good

greater stockability. Because precipitation and accuracy among data sources, both singly and

air temperature are correlated, it is difficult to aggregated by provenance (tables 4, 5, 6). Prepartition effects, which is probably why this dicted basal area growth (from [17], and [16]

analysis found temperature rather than precipi- estimated from the output of [17]) fit observed

tation (Perala et al. in review) the stronger cli- basal area growth about as well (R2=.52, sd=44

matic variable. Neither alone is satisfactory percent of mean growth) as in the model used by

because they operate at odds with each other in Schlaegel (1971) on a locally restricted subset of

the water balance and they operate in concert in these data (R2=.45, sd-37 percent).

tree physiology.

On the other hand, much local and regional bias

DeBeU et al. (1989) reported that P/nus taeda L. remains in the yield data set, and especially in

plantations in Hawaii had much greater the tabular set (tables 4, 5, 6). Although the

stockability than they did in the southeastern temperature term reduced prediction bias, the

U.S. Stockability differences were associated stockability of aspen in Newfoundland was still

with differences in tree size-class structure, greatly underestimated (table 5). Much like the

canopy depth, and leaf area index that developed Hawaii site mentioned earlier (Harms et aL 1994),

under the influence of the contrasting local Newfoundland has cool summers (July temperaclimates (Harms et al. 1994). The prime south- tures barely average 16 C) and as much as 1,200

eastern site, near Union, South Carolina, is mm precipitation annually, half again as much

much warmer and somewhat wetter (25.6 C as any of our other data sources (Page 1972,

mean July temperature and 1,247 mm annual Pearse et aL 1985). Such a climate provides a

precipitation) than the Maui, Hawaii, site (15.6 C most favorable water balance for aspen growth

and 1,143 mm). Canopies in Hawaii were 4 m to (Sucoff 1982). A water availability term (e.g.,

7 m deeper than in South Carolina, and leaf area Grier and Running 1977) in addition to, or in

index was five times greater. The favorable place of, the temperature term might aid stocking

driving influences of the Hawaiian climate were predictability.

thought to be long growing season, high solar

radiation, high sun angle, and favorable tempera- When stocking bias is examined for trends across

tures. Given that aspen regenerates greater d.b.h, classes (table 5), and d.b.h, bias is examdensities in warm environments (Schier et al. ined over age classes (table 6), five different

1985) where its life is shorter, a significant patterns among provenances emerge:

temperature term is not surprising.

1. littlebiastrendforeitherstockingor

Diameter, in contrast, increases by 7 percent per d.b.h. (Alaska, Saskatchewan, Minnedegree warming up to 17 C. Warm growing sota);

seasons should affect apical and cambial activity 2. stocking bias negative power, d.b.h, bias

proportionately (Fraser 1962, Jones and Schier positive power (Manitoba, Ontario, New1985) so that most of the temperature effect foundland, P. grandiclentata);

would be inherent in site index. Given that 3. stocking bias negative power, no d.b.h.

stockability diminishes by 18 percent per 1 C, trend (Alberta);

the inverse relationship (c.f. [4] and [5]) implies 4. stocking bias convex, d.b.h, bias concave

that d.b.h, should increase 22 percent (1/ (1-. 18)) (Rocky Mountains, Michigan);

per 1 C rather than 7 percent. This 15-point 5. stocking bias convex, d.b.h, bias positive

deficiency is added evidence that radial growth is power (P. tremu/a).

13
Table 4.--Pattem analyses of equations [8],[10], [16], and [17] by provenance and author

Author Equations [10], [17] (diameter) Equations [8], [16] (trees)

Data Cases Mean Bias Accuracy Mean Bias Accuracy

set =

mm - - - Percent- - - n/ha - - - Percent- - -

Alaska

Gregoryand Haack 1965 T 37 107 36 36 6,377 9 9

Saskatchewan

Bella1975 Y 6 11 26 26 47,060 10 12

Bellaand DeFranceschi1980 T 15 53 24 24 16,630 -12 18

Kirbyet al. 1957 T 27 149 20 20 2,619 20 20

Steneker1969 D 21 20 3 3 408 -3 4

Steneker1969 Y 10 57 -3 4 7,246 9 19 ,

All Y 16 40 8 12 22,170 9 16

All T 42 115 22 22 7,624 9 19

Manitoba

Pike1953 D 3 60 -1 6 1,523 12 12

Pike1953 Y 4 98 22 22 3,772 7 7

Steneker 1974 D 51 30 0 5 478 2 7

Steneker1974 Y 29 80 -2 11 6,339 -5 15

All D 54 31 0 5 536 2 7

All Y 33 82 1 12 6,028 -4 14

Alberta

Bella1975 Y 3 12 19 19 48,110 3 5

Bellaand DeFranceschi1980 T 18 45 28 28 19,930 4 23

Rocky Mountains '

Baker1925 T 5 287 22 22 642 3 3

BartosandLester1984 Y 3 162 -16 16 2,900 -41 41

Crouch1981 Y 1 200 -13 13 " 1,312 -4 4

JonesandTrujillo1975 Y 6 44 18 18 19,690 -23 32

KempermanandBarnes1976 Y 2 254 14 14 1,327 -62 62

Schier1975 Y 5 188 6 10 1,135 23 23

Schierand Smith 1979 Y 1 212 2 2 907 4 4

Waiters et aL 1982 Y 3 250 -27 27 927 -18 18

All Y 21 160 1 16 6,674 -15 29

.

Minnesota

Brownand Gevorkiantz1934 T 27 170 4 9 2,137 7 8

Hubbard1972 D 6 27 5 9 852 -5 7

Hubbard1972 Y 4 64 -8 9 5,651 5 11

Noreen 1968 D 5 30 1 7 441 -2 5

Noreen1968 Y 9 36 5 22 17,420 3 12

Perala1974 D 7 25 -11 11 4,806 -7 8

Perala1978 D 48 26 -2 3 128 3 5

Perala1978 Y 7 114 -5 9 3,533 3 5

Perala1979 Y 2 21 2 4 29,450 -29 34

F£able 4 continued on next page)

14
(Table 4 continued)

Author Equations [10], [17] (diameter) Equations [8], [16] (trees)

Data Cases Mean Bias Accuracy Mean Bias Accuracy

set a

mm - - -Percent- -. n/ha - - -Percent- - -

Perala1984 Y 2 50 -10 10 9,345 -21 21

Peraladatafiled D 19 19 0 3 394 2 5

Peraladatafiled Y 6 88 11 11 3,929 15 15

PeralaandAlban1982 Y 2 209 -23 23 1,022 -6 11

PeralaandLaidly1989 D 24 12 1 3 20 1 2

Schlaegel1971,1972 D 60 27 0 2 89 7 7

Schlaegel1975 Y 1 182 -20 20 1,334 -17 17

SchlaegelandRingold1971 D 8 26 0 2 31 5 5

All D 177 24 0 3 343 3 6

All Y 33 80 -1 14 9,354 1 13

Michigan

Barnes1969 Y 3 86 6 7 2,723 45 45

Day1958 D 3 20 6 6 82 3 5

Day 1958 Y 7 56 7 8 8,566 -6 17

Graham et al. 1963 T 16 171 -26 26 961 51 51

All Y 10 65 7 8 6,813 9 26

Ontario

Plonski1960 T 51 190 1 10 1,337 18 18

Newfoundland

Page 1972 T 8 137 15 15 3,614 -29 34

Europe (P. tremula and hybrid)

Elfving1986 D 4 58 -2 11 55 -6 7

Haugberg1958 D 69 15 1 2 <1 -1 1

Haugberg1958 Y 34 144 -2 15 1,862 19 27

Vincent et al. 1950 T 50 162 0 20 2,227 10 20

Vuokila1977 D 11 22 1 3 34 -3 3

Vuokila1977 Y 2 40 21 21 12,950 16 24

All D 84 18 1 2 7 -2 2

All Y 36 139 -1 15 2,478 19 27

TOTAL P. tremuloides, tremula D 339 23 0 4 1 5

TOTAL P. tremuloides, tremula Y 152 99 2 14 3 20

P. grandidentata

Graham et al. 1963 T 19 184 -28 28 936 24 27

Koerperand Richardson1980 Y 3 166 21 21 1,474 -1 18

aT = tables,Y = yield,D = dynamic.

15
Table 5.--Pattem analyses of equations [8] and [16] by quadratic mean d.b.11, class, data set, and

provenance

Quadratic mean d.b.h, class, cm

Provenance Cases <5 5 10 15 20 25 >25 Mean

............ Bias,percent ............

Dynamic data set (eq. [16])

Saskatchewan 21 -9 -2 -1 -3

Manitoba 54 2 3 2 1 -2 2

Minnesota 177 -3 2 6 5 6 5 3

Michigan 3 3 3

P. tremula 84 -7 -4 -2 -1 -1 -2 -2

MEAN -3 1 2 2 4 4 1

CASES 39 83 80 71 45 21 339

Yield data set (eq. [8])

Saskatchewan 16 10 13 -7 9

Manitoba 33 4 -5 -20 -4

Alberta 3 3 3

RockyMountains 21 -23 -17 -2 -11 -36 -15

Minnesota 33 -16 7 -2 4 -11 5 1

Michigan 10 -13 42 9

P. tremul_ 36 7 18 17 44 _ 19

MEAN 0 2 7 5 10 -7 -22 3

CASES 13 53 37 29 13 4 3 152

Tabular data set (eq. [8])

Alaska 37 8 9 9 8 8 9

Saskatchewan 42 10 8 -3 15 20 14 12 9

Alberta 18 29 -13 -26 4

Rocky Mountains 5 4 3 3

Minnesota 27 17 9 6 6 3 2 7

Michigan 16 60 56 46 37 32 51

Ontario 51 60 31 17 15 9 9 18

Newfoundland 8 10 -16 -41 -56 -29

P. tremula 50 11 21 14 7 0 -5 10

MEAN 22 9 15 14 12 8 4 12

CASES 13 44 61 45 42 27 22 254

P. grandidentata data set (eq. [8])

Michigan 22 56 24 9 15 20

16
Table 6._Pattem analyses of equations [10] and [17] by age class, data set, and provenance

Age class_years

Provenance Cases <10 10 20 30 40 50 >50 Mean

............ Bias,percent ............

Dynamic data set (eq. [17])

Saskatchewan 21 3 2 3

Manitoba 54 -4 -1 2 -1 0

Minnesota 177 0 -2 0 0 1 0 0

Michigan 3 6 6

P. tremula 84 -17 -1 5 1 0 0 1

MEAN -1 -1 0 1 1 0 0

CASES 26 75 56 77 70 35 339

Yield data set (eq. [10])

Saskatchewan 16 44 -1 1 1 8

Manitoba 33 -8 1 -3 11 16 1

Alberta 3 19 19

RockyMountains 21 18 -23 -4 1

Minnesota 33 24 -3 4 0 -12 -27 -1

Michigan 10 13 5 7 7

P. tremula 36 22 -11 -2 -3 9 -6 -1

MEAN 29 -2 5 -1 0 -1 -3 1

CASES 8 32 36 12 39 7 18 152

Tabular data set (eq. [10])

Alaska 37 22 40 37 35 38 36

Saskatchewan 42 18 18 29 30 21 18 22

Alberta 18 29 24 27 32 28

RockyMountains 5 22 22

Minnesota 27 1 7 11 9 0 4

Michigan 16 -21 -26 -34 -24 -26

Ontario 51 -16 -11 -6 -3 8 1

Newfoundland 8 8 14 11 9 20 15

P. tremula 50 -54 -19 -11 -(_ 1 17 0

MEAN 12 0 9 8 7 17 11

CASES 18 34 32 38 26 106 254

P.grand/dentata data set (eq. [10])

Michigan 22 -50 -41 -32 -5 3 -21

17
Such variability indicates additional environmen- Maritimes, and the warmest extremities of the

tal or genetic control of both the rate and power aspen range.

parameters of the self-thinning and growth-age

relationships beyond our ability to explain with ACKNOWLEDGMENTS

the given data base. Developing local equations

or accepting variability as noise on a global scale We thank I. Bella, R. McRoberts, and T. Mowrer

are the only altematives until we can develop for their helpful reviews of an earlier draft. We

ecophysiological models (Rauscher et al. 1990) dedicate this paper to Bryce E. Schlaegel, who

that better reflect the growing environment, enlightened our understanding of aspen producAlthough local genetic variability can be over- tivity. He died much too young in Atlanta,

whelming (Jones and DeByle 1985), it is not Georgia, in March 1990.

necessarily a deterrent. For example, three of

the studies in this analysis are on the same soft LITERATIm_ CITED

series, Warba fine sandy loam (Perala and Alban

1982), but as much as 8 km apart (Schlaegel Anon. 1984. Climatic atlas climatique -

1971; Perala 1974, 1978). The overall accuracy Canada. Environment Canada, Atmospheric

was 4 percent for diameter estimation and 5 Environment Service. Map Series 1 - Tempercent for stockability, despite the many clones perature and degree days. Unpaged.

included. (1951-1980).

Stockabiltty of largetooth aspen was significantly Anon. 1986. Climatic atlas climatique -

greater than quaking aspen in the same locale Canada. Environment Canada, Atmospheric

(Graham et al. 1963 in table 5). The Eurasian Environment Service. Map Series 2 - Precipiaspens did not clearly differ from quaking aspen, tation. Unpaged. (1951-1980).

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aspen. Certainly, aspen conforms to delineation of clones of Populus

self-thinning relationships, but the rate param- tremuIo/de$ and P, grandfdentata in northeter and perhaps even the power parameter ern lower Michigan. Silvae Genetica. 18(4):

appear to be driven in part by the local growing 97-144.

environment. Genetic variability at the clone

level, beyond that expressed as measured site Bartos, D.L.; Lester, J.E. 1984. Effects of 2,4-D

index, likely affects these coefficients as well. on a Populu$ tremu/o/de$ commllnity in

Other sources of variation might include study the western United States---22 years after

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Examines the d.b.h.-age and stockability relationships in aspen

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trend in the self-thinning relationship in young aspen stands.

KEY WORDS: survival, self-thinning, size-density, limiting relationship.