United States

Department of

Agriculture

Forest Service

Evaluation of Two Eastern White

p@ hq

0

asalar

Southern

Research Station

Pine Site Index Equations at

Biltmore Estate, North Carolina

W. Henry McNab, Bernard R. Parresol,

and Brian A. Ritter

Research Paper

SRS-21
The Authors

W. Henry McNab is a research forester with the Ecology and

Management of Southern Appalachian Hardwoods Research Work Unit,

U.S. Department of Agriculture, Forest Service, Southern Research

Station, Asheville, NC 28806; Bernard R Parresol is a mathematical

statistician with the Southern Forest Inventory, Monitoring, and

Analysis Program (SFIMAP), U.S. Department of Agriculture, Forest

Service, Southern Research Station, Asheville, NC 28804; and Brian A.

Bitter is the forestry supervisor with the Biltmore Estate, Asheville, NC

28801.

Cover: Eastern white pines 106 years from seed on plot la of the Old Orchard Plantation, Biltmore Estate,

in August 2000. Individually identified trees on this and two other study plots have been measured

periodically for diameter andheight since October 1928, at 34 years.

August 2000

Southern Research Station

P.O. Box 2680

Asheville, NC 28802
Evaluation of Two Eastern White Pine

Site Index Equations at Biltmore Estate,

North Carolina

W. Henry McNab, Bernard R. Parresol, and Brian A. Ritter

Abstract

The accuracy of two white pine (Pinus strobus L.) polymorphic site

index equations was compared with field data from three plots in a

loo-year-old stand at Biltmore Estate, Asheville, NC. One equation

was developed from New Hampshire data and the other from Southern

Appalachian data Tree height has been measured periodically on

those plots between stand age 34 and 72 years. At 50 years, site index

ranged from 54 to 73 feet (ft). Absolute deviations of actual from

estimated site index averaged 2.5 A for the Southern Appalachian

equation and 2.7 ft for the New Hampshire equation. Site index

estimates of the New Hampshire model were highly correlated

(r = 0.92) with estimates of the Southern Appalachian model. Our

evaluation indicates that the New Hampshire equation, which has

desirable properties lacking in the Southern Appalachian equation,

may be applicable on low- to moderate-quality sites beyond the area

in which it was to be used. We suggest that a regional approach to site

index development for white pine may be appropriate.

Keywords: Base-age invariant, height growth, invertible, Pinus sfrobus

L., polymorphic, Southern Appalachian Mountains.

Introduction

Eastern white pine (Pinus strobus L.) is a widespread

commercial timber species in the Eastern United States that

responds well to management in planted and natural, as well

as pure and mixed species stands, over a broad range of site

quality (Burns and Honkala 1990). Methods for evaluating

its site quality have long been available (Frothingham 1914),

particularly based on site index (the height reached by a

forest stand at a selected in&x age, usually 50 years)(Beck

197 lb). Parresol and Vissage (1998) developed a site index

equation for eastern white pine (hereinafter white pine) in the

Southern United States that has desirable characteristics

lacking in other models. However, it was developed with data

only from New Hampshire and was not evaluated for area of

applicability. Although a polymorphic site index model is

available for white pine in the Southern Appalachian

Mountains (Beck 197 1 b), it is not as versatile as the New

Hampshire model, nor has it been field tested using an

independent data set.

Results of a study comparing predicted with actual white

pine site index over a range of stand ages in the Southern

Appalachian Mountains are presented in this paper. Its

objectives are to compare precision of white pine site index

estimates using two models (Beck 197 1 b, Parresol and

Vissage 1998) and to compare site index estimates using the

New Hampshire model with estimates using the Southern

Appalachian model. The site index model developed by Beck

(197 1 b) has long been accepted as applicable throughout the

Southern Appalachian region and is used as a standard for

comparison of other models.

Methods

Site Index Equations

Characteristics of the two site index equations are listed in

table 1. Both are polymorphic, which permits more accurate

modeling of height growth over a wide range of site qualities

than do anamorphic equations. Invertibility facilitates

estimation of stand height by rearrangement of the site index

model form. Age invariance refers to changing the

equation’s index age without refitting the data. The Beck

(1971 b) equation is expressed only in terms of height:

H = [63 06 + 0.67(s)] { ~~~~p-~~~~~~~~~~~~~~~~~,l1)~ ,

where

H = total mean dominant height (feet),

S = site index at base age 50 years (feet), and

t = stand age (years).

Estimates of site index must be obtained iteratively by

varying stand height and age.

The Parresol and Vissage (1998) equation for base age 50 is

formulated as

Ins = l/1.881 exp(-8.6188/A)(lnI-I+ 74.7099/A

- 2.0862) + 0.5920

where

In = natural logarithm, and

A = stand age.
Table l-Characteristics of two equations developed for estimating site index of

eastern white pine (pinusstrobus L.) in the Southern Appalachian Mountains

Characteristics of site

index data and equation

Data source

Latitude range (degrees)

Longitude range (degrees)

Stands sampled (number)

Age range (years)”

Stand height (feet)

Site index range (50 yrs)

Polymorphic

Age invariant

Invertible

(1Eli)

GA, NC, TN, VA

34-3 7”

82-84”

42

44-70

n/at

71-122

YeS

No

No

Parresol and

Vissage (1998)

NH

43-45”

71-72”

196”

10-100

4-113

50-90

YE4

YeS

YeS

’ Summarized data from Frothingham (1914)

* Site index equation applicable from ages 5 to 70 years (Beck 1971b)

‘Not available

The Southern Appalachian equation was developed over a proportion are chosen. The final number of sample trees

broader range of site indices than the New Hampshire ranged from 11 to 22 per plot, equivalent to between 88 and

equation, but the Parresol and Vissage (1998) equation is 96 trees per acre. We included only those trees alive at the

applicable in stands older than 70 years. Both models are last inventory so that crown class could be verified in the

applicable in stands as young as 10 years. field as dominant or codominant.

Study Area and Data Analysis

We conducted the field portion of this study in the Old

Orchard Plantation on Biltmore Estate located near

Asheville, North Carolina. This plantation was established

with 4-year-old white pine seedlings planted at an average

spacing of 3 by 4 fi in March 1899. In the fall of 1916, when

the trees were 22 years from seed, a study was begun to

investigate the effect of stand density on growth and yield.

Three small contiguous plots: la (0.25 acre), lb (0.125 acre),

and lc (0.125 acre), each surrounded by a 0.5~chain buffer

zone, were established in a relatively uniform, 1.6-acre

portion of the 5.6-acre stand (Haasis 1930). These plots have

been the basis of periodic reports on diameter and volume

increment, most recently by Della-Bianca (198 1). Trees on

each plot were identified in 1916 and inventoried periodically

for diameter at breast height (d.b.h.) and total height at ages

34,41,47,52,58,72, and 100 years. We excluded the lOO-year

data from our tests because it exceeded the age range of the

Southern Appalachian equation. Stand ages 34 and 4 1 were

included, however, because the Southern Appalachian

model is applicable at younger ages (Beck 197 1 b).

Heights of 11 and 16 sample trees were missing at ages 34

and 72 years, respectively; these were estimated using 4-

point Lagrangian interpolation. The Lagrange polynomial

through 4 points is

v - X,MX - KMX - X3) (X - W(X - W(X - X3)

Y=Yo +y,

(xl - X,M% - %uil - J&J (X, - TAX, - wx, - q+

(X - J&,)(X - X,)(X - J&3,) (X - &ax- x,xX - w

Y2 + Y3

(x, - WV2 - X,Mx, - $1 w3 - %m- wcx, - x2)

where

Y = interpolated tree height,

.X = age at which Y is to be interpolated, and

Y,,, Y,, Y2, Y, = respective heights at ages &, X,, &, X,

We selected sample trees using a method designed by Zeide

and Zakrzewski ( 1993) in which a fixed number and a fixed

For details on Lagrangian interpolation see Pachner (1984).

Lagrangian interpolation was also used to estimate tree

height at 50 years for determination of measured site index.

We used the 95-percent confidence limits of measured site

indices to assess the estimated values the two equations

produced With field data t?om each of the six inventories,

2
we used each equation to calculate site index. We then

determined site index deviations (actual - estimated) at each

periodic inventory for each equation. Predicted site index

was then compared with the actual value determined at 50

years. Precision of the two equations was evaluated with

percent standard error (Schlaegell98 1), using differences of

actual - estimated site index at each of the six periodic

inventories. The percent standard error is calculated as

actual site index - estimated site index

S(%) = x100.

estimated site index

This statistic indicates the size of error as a percentage of

the mean of the site index distribution. Plotting and

regression were used to further evaluate the relationship

between site index predicted by the two equations. We used

simple regression analysis to compare the relationship of site

index estimated with the New Hampshire model to site index

estimated with the Southern Appalachian model. In

addressing the precision of his site index curves, Beck

( 197 1 b) wrote: “There was no evidence of bias at any age,

and the accuracy expected in even the lo-year-old stands is

probably sufftcient for most management purposes.”

Results and Discussion

Height growth of trees on all plots in the Old Orchard

Plantation was nearly linear between ages 34 and 72 years

(fig. 1). Annual increment ranged from 1 .O ft on plot 1 c to 1.5

ft on plots la and lb. The overall average of 1.38 ft is slightly

greater than reported by Beck (197 la) for trees on lowquality sites in the Southern Appalachians. At 50 years, total

heights of sample trees on the three plots averaged 53.8,

66.l,and73.48.

We rounded mean sample tree height on each plot to the

nearest foot to obtain site index; e.g., 54,66, and 73 ft (table

2). These values closely agree with previous estimates of 56,

7 1, and 74 ft that were calculated from the tallest seven trees

per 0.125-acre plot (Della-Bianca 1981). The standard

deviation of heights at age 50 was greatest (7.8 fi) in plot la,

which is situated between plots 1 b and 1 c and is more

variable in site quality. These site in&x values are within the

range of curves presented by Parresol and Vissage (1998),

but not for the Southern Appalachian equation, which is

limited to a minimum site index of 60 ft. To facilitate

comparisons, we extrapolated Beck’s (197 1 b) equation to site

index 54.

8 0

6 0

5 0

4 0

30 40 50 60 70

Age (Yew

Figure l-Total height and age relationships of eastern white pines on

three plots in the Old Orchard Plantation, Biltmore Estate, Asheville,

NC. Plotting symbols indicate average height of the dominant and

codominant stand at inventory years, except at age 50 years when

height was interpolated.

Table 2-Characteristics of dominant and codominant

eastern white pine stands interpolated at 50 years of age for

determination of site index on three plotx in the Old

Orchard Plantation, Biltmore Estate, Asheville, NC

statistic Plot lc Plot la Plot 1 b

Plot area (acre) 0.125 0.25 0.125

Trees sampled (number) 12 22 1 1

Age from seed (yrs) 50 50 50

Mean stand height (fty 53.8 66.1 73.4

Minimum tree height (tt)(l 47.8 47.7 66.9

Maximum tree height (8)’ 62.4 78.0 81.7

Standard deviation (tt) 4.8 7.8 4.6

Standard error (fl) 1.4 1.6 1.4

a Determined by Lagrangian interpolation

3
---a---- Beck (197 1 b) ---m--- Parresol and Vissage (1998)

Plot lb

60 IIIIIIIII~IIIIIIIIIIII

7 0 Plot lc

30 40 50 60 70 80

Plot la

7 0

.._.__.___.__._.

6 0

5 0

40 

3 0 4 0 50 60 7 0 8 0

Figure 2-Precision of white pine site index estimation at periodic inventories using equations developed by Beck

(1971b) and Parresol and Vissage (1998) on three sample plots in the Old Orchard Plantation, Biltmore Estate,

Asheville, NC. The range of total heights of sample trees interpolated at 50 years of age on each plot is indicated by

length of the vertical line. The two parallel, horizontal, dashed lines delineate the 95-percent confidence interval of

mean site index.

The pattern of estimated to actual site index was consistent

for both equations on all plots (fig. 2). Site index estimates of

both equations were within the 9%percent confidence limits

between years 40 and about 65. The Southern Appalachian

model was particularly ill-behaved beyond age 60 when

estimates of site index abruptly increased and crossed

estimates of the New Hampshire equation. Mean deviations

of site index were greatest (2.8 ft) in plot lc, and deviations

were least (2.2 fi) in plot lb (table 3). For all plots combined,

mean absolute deviations of site index were about equal for

the Southern Appalachian equation (2.53 fit) and the New

‘Hampshire equation (2.67 tt). One difference between the

two equations is that stand height is equivalent to site index

at the base age of 50 years for the New Hampshire equation

but not for the Southern Appalachian equation. Estimated

site index based on Beck’s (197 1 b) model averages 2.7

percent less than tree height at the base age.

Comparison of standard errors for stand ages 34 to 72 years

indicates no clear trend of site index estimates between the

two equations (table 3). Maximum standard errors occurred

at the youngest plantation age (34 years) for the New

Hampshire equation but at the oldest (72 years) for the

Southern Appalachian equation. Errors were smallest for plot

lb in both equations. For young stand ages (30 to 40 years),

where estimates of site index are important for estimating

future yield, Beck’s (1971b) equation is most accurate,

although at 34 years neither equation provided satisfactory

results. For estimates beyond 40 years, errors were smallest

with the New Hampshire equation. Users should remember

that our tests were conducted in three small plots in a single

4
Table 3-Average total height, deviation, and percent standard error of two site index equations applied

at periodic inventories in three plots of the Old Orchard White Pine Plantation, Biltmore Estate,

Asheville,NC

Plot numbers Age

Beck(l97lb) Parresol and Vissage (1998)

Height (Dev.) S(%yl (Dev.) S(%)

lc 34 41.4

lc 4 1 47.0

lc 47 50.8

lc 5 2 55.8

lc 5 8 60.2

lc 72 70.3

Mean

Absolute mean

la 34 50.3

la 4 1 58.0

la 47 63.4

la 5 2 68.0

la 5 8 73.9

la 7 2 91.6

Mean

Absolute mean

lb 34 55.5

lb 4 1 65.1

lb 47 70.3

lb 5 2 75.5

lb 58 81.2

lb 7 2 96.8

Mean

Absolute mean

Overall mean

Absolute overall mean

Yr.9 F t

-5.7 -9.5 -8.0 -12.9

-1.5 -2.7 -3.1 -5.4

1.7 3.3 0.1 0.1

1.9 3.6 0.1 0.2

3.0 5.9 1.0 1.8

3.0 -5.6 1.0 2.0

0.4 -0.8 -1.5 -2.4

2.8 5.1 2.2 3.7

-4.4 -6.3 -8.2 -11.1

-1.5 -2.2 -3.9 -5.6

0.9 1.4 -1.0 -1.5

1.8 2.8 0.3 0.4

1.9 3.0 0.6 0.9

-5.2 -11.1 -4.0 -5.7

-1.1 -2.1 -2.7 -3.8

2.6 4.5 3.0 4.2

-3.5 -9.0 -8.3 -10.2

-2.0 -5.3 -5.1 -6.6

0.9 -0.3 -1.2 -1.6

1.4 1.3 0.0 -0.1

1.7 2.7 0.9 1.3

-3.5 -0.3 -1.2 -1.6

-0.8 -1.8 -2.5 -3.1

2.2 3.2 2.8 3.6

-0.50 -1.57 -2.22 -3.09

2.53 4.24 2.67 3.83

’ S(%) = [(actual - estimated)/estimated] 100
stand on a site of lower productivity *&an is typical of the

Southern Appalachians. Our results should not be

extrapolated to other areas without first conducting

additional validation tests.

youngest and oldest ages. According to our regression

analysis, the New Hampshire model expresses a relationship

similar to that of the Southern Appalachian model between

site indices 5 1 to 76 (regression slope = 0.98), but estimates

of the New Hampshire model slightly exceed those of the

Site in&x estimated using the New Hampshire model was Southern Appalachian model (intercept = 2.1 ft). The close

highly correlated (r* = 0.92) with the Southern Appalachian agreement of site index estimates between the two models

model (fig. 3). We observed that the greatest deviation suggests that development of a single regional equation for

tended to be associated with site index estimates at the white pine may be feasible.

8 5

8 0

6 5

6 0

Parresol and Vissage (1998) =

_ 2.09 + 0.985* (Beck 1971b)

/’

/’ /’ 1’ /’ /’

5 0 5 5 60 65 7 0 7 5 8 0 8 5

Site index (Beck 197 1 b)

Figure 3-Site index estimates of eastern white pine determined by Beck (1971b) and Parresol and Vissage (1998) in the Old

Orchard Plantation, Biltmore Estate, Asheville, NC. The dashed line indicates perfect correlation between the two models.

The solid line represents the least squares linear regression for the New Hampshire model as a function of site index

estimated by the accepted standard model (Beck 1971b).
Several explanations may be relevant for the results of this

study. First, tree height growth does not exhibit the typical

sigmoid shape assumed by the equations. The height

increment of white pine stands on low-quality sites such as

the Old Orchard is typically linear (Beck 197 1 a); but neither

site index equation adequately described relationships

among the three plots. Second, both equations were

developed with data from natural stands but were applied in

a stand that had been planted and a portion of which had

been thinned. The effects of vegetation control on early

height growth during white pine stand development are

unknown. Third the effects of thinning (plot 1 a) on white

pine height increment are unknown. Lastly, at very young

and old stand ages, the different equation formulations used

in the two models likely account for some ofthe lack of

agreement between the two site index equations. These

variable vegetative responses may cause users to question

site index as an ideal measure of site quality (Beck and

Trousdell973).

In summary, the two equations performed similarly on three

relatively low-quality sites in one stand in the Southern

Appalachian Mountains. However, neither accurately

depicts the linear pattern of white pine height increment on

the small and unreplicated plots of the Old Orchard stand.

Estimates of actual site index were slightly more accurate

using the Southern Appalachian equation for stand ages

less than 40, but they were more accurate using the New

Hampshire equation in stands 40 years and older. We

suggest that it is desirable and feasible to develop a regional

site index equation for white pine.

Acknowledgments

Helpful suggestions for data analysis and summarization of

site index comparisons were provided by F. Thomas Lloyd

and are gratefully acknowledged We also thank William B.

Leak and three anonymous reviewers for constructive

suggestions.

Beck, D.E. 197la. Height-growth patterns and site index of white

pine in the Southern Appalachians. Forest Science. 17: 252-260.

Beck, D.E. 197lb. Polymorphic site index curves for white pine in

the Southern Appalachians. Res. Pap. SE-80. Asheville, NC: U.S.

Department of Agriculture, Forest Service, Southeastern Forest

Experiment Station. 8 p.

Beck, D.E.; Troosdell, K.B. 1973. Site index: accuracy of

prediction. Res. Pap. SE-108. Asheville, NC: U.S. Department of

Agriculture, Forest Service, Southeastern Forest Experiment

Station. 7 p.

Burns, R.M.; Honkaln, B.H. 1990. Silvics of North America.

Conifers. Agric. Handb. 654. Washington, DC: U.S. Department of

Agriculture. 675 p. Vol. 1.

DeBa-Biancr, L. 1981. Thinning eastern white pine in the Southern

Appalachians. Southern Journal of Applied Forestry. 5: 68-73.

Frothigham, E.H. 1914. White pine under forest management.

Agric. Bull. 13. Washington, DC: U.S. Department of Agriculture.

70 p.

Hansis, F.W. 1930. Forest plantations at Biltmore, North Carolina.

Misc. Pub. 61. Washington, DC: U.S. Department of Agriculture.

29 p.

Pachner, J. 1984. Handbook of numerical analysis applications. New

York: McGraw-Hill Book Company. 512 p.

Parresol, B.R.; Vissage, J.S. 1998. White pine site index for the

southern forest survey. Res. Pap. SRS-10. Asheville, NC: U.S.

Department of Agriculture, Forest Service, Southern Research

Station. 8 p.

Schlaegel, B.E. 1981. Testing, reporting, and using biomass

estimation models. In: Gresham, C.A., ed. Proceedings of the 1981

southern forest biomass workshop; 1981 June 11-12; Georgetown,

SC: The Belle W. Baruch Forest Science Institute of Clemson

University: 95-l 12.

Zeide, B.; Zakrzewski, W.T. 1993. Selection of site trees: the

combined method and its application. Canadian Journal of Forest

Research. 23: 1019-1025.

*U.S. -PRlINl'IK OFFICE: 2000~565-019/00032

7

McNab, W. Henry; Parresol, Bernard R; Ritter, Brian A. 2000. Evaluation of two eastern

white pine site index equations at Biltmore Estate, North Carolina. Res. Pap. SRS-21.

Asheville, NC: U.S. Department of Agriculture, Forest Service, Southern Research Station.

P *

The accuracy of two white pine (Pinus strobus L.) polymorphic site index equations was

compared with field data from three plots in a loo-year-old stand at Biltmore Estate, Asheville,

NC. One equation was developed from New Hampshire data and the other from Southern

Appalachian data. Tree height has been measured periodically on those plots between stand age

34 and 72 years. At 50 years, site index ranged from 54 to 73 feet (ft). Absolute deviations of

actual from estimated site index averaged 2.5 ft for the Southern Appalachian equation and 2.7 ft

for the New Hampshire equation. Site index estimates ofthe New Hampshire model were highly

correlated (r = 0.92) with estimates of the Southern Appalachian model. Our evaluation indicates

that the New Hampshire equation, which has desirable properties lacking in the Southern

Appalachian equation, may be applicable on low- to moderate-quality sites beyond the area in

which it was to be used. We suggest that a regional approach to site index development for white

pine may be appropriate.

Keywords: Base-age invariant, height growth, invertible, Pinus strobus L., polymorphic, Southern

Appalachian Mountains.
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