Growth Models and Profile Equations for Exotic Tallow Tree (Triadica Sebifera) in Coastal

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Growth models and profile equations for exotic tallow tree (Triadica sebifera) in coastal

Mississippi

Nana Tian

A Thesis Submitted to the Faculty of Mississippi State University in Partial Fulfillment of the Requirements for the Degree of Master of Science in Forestry in the Department of Forestry

Mississippi State, Mississippi

May 2014

Nana Tian

2014

Growth models and profile equations for exotic tallow tree (Triadica sebifera) in coastal

Mississippi

Nana Tian

Approved:

______Zhaofei (Joseph) Fan (Major Professor)

______Thomas G. Matney (Committee Member)

______Emily B. Schultz (Committee Member)

______Michael Darragh Ulyshen (Committee Member)

______Andrew W. Ezell (Graduate Coordinator)

______George M. Hooper Dean College of Forest Resources

Name: Nana Tian

Date of Degree: May 16, 2014

Institution: Mississippi State University

Major Field: Forestry

Major Professor: Zhaofei Fan

Title of Study: Growth models and profile equations for exotic tallow tree (Triadica sebifera) in coastal Mississippi

Pages in Study: 67

Candidate for Degree of Master of Science

Chinese tallow tree (Triadica sebifera (L.)) has become a threat to Southern

United States forestlands because of the rapid colonization. To explore the growth rate of tallow, numerous growth models were constructed with destructive sampling method from oak-gum-cypress (Quercus/Liquidambar styraciflua/Taxodium distichum) and

longleaf/slash pine (Pinus taeda/ Pinus echinata) forests in southern Mississippi.

Moreover, stem profile was also fitted with segmented profile models: Max and Burkhart

(1976), Cao (2009) modified Max and Burkhart, and Clark et al. (1991). Numerous

results showed that: 1) diameter at breast height, volume and biomass of tallow grew

faster in oak-gum-cypress forest while height grew faster with DBH in Longleaf/Slash

pine forest; 2) the stem of tallow was generally sturdy in oak-gum-cypress forest while it

was slender in longleaf/slash pine forest; however, there was no significant difference

between them. Growth and taper models provide a tool for managers to estimate future

stocking of tallow tree.

ACKNOWLEDGEMENTS

I am appreciate all the support and help I received from my supervisor Dr.

Zhaofei Fan. Under his guidance, I finished this research as planned. Moreover, I learned his rigorous attitude of scholarship and hard-working spirit. I would also like to express my gratitude to Dr. Emily Schultz and Dr. Thomas Matney for their assistance and comments during my study. I am grateful for Dr. Michael D. Ulyshen at the U.S. Forest

Service Wood Products Insect Research Lab (Starkville, MS), for his patient help and direction during all field data collection.

In addition, I wish to acknowledge my lab mates (Allison Stoklosa, Mary F.

Nieminen, Carol Sui, Michael K. Crosby, and Dr. Yanping Wang) and Craig Bell and

Shawn Cooper for their help in collecting data, and I would like to thank the landowners who made my work possible. Finally, I am grateful for my parents for their support, love, and valuable education.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... ii

LIST OF TABLES ...... v

LIST OF FIGURES ...... vii

CHAPTER

I. INTRODUCTION ...... 1

II. LITERATURE REVIEW ...... 3

2.1 Studies on exotic tallow tree ...... 3 2.2 Growth models of tree species ...... 5 2.3 Tree profile functions ...... 7

III. METHODS ...... 9

3.1 Data Collection ...... 9 3.2 Model Development ...... 13 3.2.1 Height, Diameter, and Volume/Biomass equations ...... 13 3.2.2 Taper Functions ...... 15 3.3 Models Evaluation ...... 17

IV. RESULTS ...... 19

4.1 Sites effects test...... 19 4.2 Fitted Models ...... 24 4.2.1 Diameter, Height, and Volume equations ...... 24 4.2.2 Taper Functions ...... 27 4.2.2.1 Model Parameters ...... 27 4.2.2.2 Comparison of Taper Models ...... 30

V. DISCUSSION AND COLUSION ...... 50

5.1 Discussion ...... 50 5.2 Conclusions ...... 53

iii

REFERENCES ...... 55

APPENDIX

A. FIELD AND LAB IMAGES ...... 61

LIST OF TABLES

3.1 Descriptive statistics for sample tallow trees in Poplarville, Grand Bay, and Sandhill, MS ...... 12

4.1 SAS results for DBH, height, volume, biomass growth models and profile functions for tallow tree at Poplarville, Grand Bay, and Sandhill (the reference point), MS, United States...... 20

4.2 Parameters estimated for DBH, height, and volume/biomass growth models and allometric relationship models among them ...... 26

4.3 Estimated parameters and fit statistics for the Max and Burkhart profile model for tallow at Poplarville, Grand Bay, and Sandhill, MS, sites...... 27

4.4 Estimated parameters and fit statistics of the Cao (2009) profile model for tallow at Poplarville, Grand Bay and Sandhill, MS, sites...... 28

4.5 Estimated parameters and fit statistics of the Clark, Souter and Schlaegel profile model for tallow at Poplarville, Grand Bay and Sandhill, MS, sites...... 28

4.6 RMSE and FI for the estimated diameter outside bark (DOB) profiles of tallow at Poplarville and Grand Bay/Sandhill, MS, sites ...... 32

4.7 RMSE and FI for the estimated diameter inside bark (DIB) profiles of tallow at Poplarville and Grand Bay/Sandhill, MS, sites ...... 32

4.8 Residuals, RMSE, and bias of Max and Burkhart taper model for DOB at Poplarville, MS...... 34

4.9 Residuals, RMSE, and bias of Cao (2009) model for DOB at Poplarville, MS...... 35

4.10 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DOB at Poplarville, MS...... 36

4.11 Residuals, RMSE, and bias of Max and Burkhart taper model for DOB at GrandHill, MS...... 37

4.12 Residuals, RMSE, and bias of Cao (2009) profile model for tallow DOB at GrandHill, MS...... 38

4.13 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DOB at GrandHill, MS...... 39

4.14 Residuals, RMSE, and bias of Max and Burkhart taper model for DIB at Poplarville, MS...... 40

4.15 Residuals, RMSE, and bias of Cao (2009) model for tallow DIB at Poplarville, MS...... 41

4.16 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DIB at Poplarville, MS...... 42

4.17 Residuals, RMSE, and bias of Max and Burkhart taper model for DIB at GrandHill, MS...... 43

4.18 Residuals, RMSE, and bias of Cao (2009) profile model for tallow DIB at GrandHill, MS...... 44

4.19 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DIB at GrandHill, MS...... 45

LIST OF FIGURES

3.1 Flow chart outlying experiment methods in the field and laboratory ...... 11

3.2 Disk abstraction (circle) and section separation (slash) along tree stem ...... 11

3.3 Boxplot for DBH, height, volume, and biomass collected from Poplarville (Poplar), Grand Bay (Grandbay), and Sandhill sites ...... 13

4.1 Height/DBH, DBH/age, height/age, and volume/age trends based on observations of tallow at Poplarville (blue dots), Grand Bay NERR (green diamonds), and Sandhill NWR (red stars) ...... 21

4.2 Biomass/age, volume/DBH, and volume/biomass/DBH2*height trends based on observations of tallow at Poplarville (blue dots), Grand Bay NERR (green diamonds), and Sandhill NWR (red stars) ...... 22

4.3 DOB (A and C) and DIB (B and D) stem profile trends based on measurements of tallow at Poplarville, Grand Bay, and Sandhill, MS, sample sites...... 23

4.4 Fitted DOB stems profiles for tallow tree at Poplarville MS (A) and Grand Bay/Sandhill MS (B) sites using Max and Burkhart (dash), Cao (2009) (round dot), Calrk et al. (1991) (solid) ...... 29

4.5 Fitted DIB stem profiles for tallow tree at Poplarville MS (C) and Grand Bay/Sandhill MS (D) sites using Max and Burkhart (dash), Cao (2009) (round dot), Calrk et al. (1991) (solid) ...... 30

4.6 Comparison of the residuals of three models for DOB profile function at Poplarville site (A) and Grand Bay/Sandhill (B), MS, test sites...... 46

4.7 Comparison of the bias of the three models for DIB profile function at Poplarville site (a) and at Grand Bay/Sandhill (b), MS, test sites...... 47

4.8 Best fitted stem profile model for both DOB (A) and DIB (B) of tallow tree at Polplarville (dash) and Grand Bay/Sandhill (solid), MS, respectively ...... 49

A.1 Fell down the sample tallow trees in sample sites ...... 62

vii

A.2 Separate the tree stem into 1 meter sections and abstract disks from the middle of each section ...... 63

A.3 Measure the volume of each disk with water displace method ...... 64

A.4 The sander used to sand all disks before counting rings ...... 65

A.5 Method of sanding all disks with the equipment before getting age of each tree ...... 66

A.6 Counting rings on each disk with pins marking them...... 67

viii

CHAPTER I

INTRODUCTION

The Chinese tallow tree (Triadica sebifera (L.)) in the spurge family

(Euphorbiaceae) is a monoecious, deciduous tree, native to central and southern China

(Zhang et al., 1994). Since its introduction as an ornamental and potential oil producing species in the 1770s, it has become an invasive species in the southern forestlands of the

United States (Bruce, 1993). Previous study recorded that there were approximately

185,000 acres of southern forests had been invaded by tallow tree (Tan, 2012) especially the coastal prairies and plains (Gan et al., 2009). For example, tallow invaded along the

Gulf Coast of the U.S. where forestlands are impacted by land falling hurricanes

(Chapman et al., 2008). Currently, tallow is also present along the East Coast and is becoming a dominate species there (Bruce et al., 1997; Pattison and Mack, 2009). To date, it has invaded nine southern states and the most severe occupations are in

Mississippi, Louisiana, Alabama, and Texas (Miller et al., 2008; Fan et al., 2012).

Tallow is very harmful to the southern forestlands and has caused severe ecological and economic loss to native ecosystems. It is referred to as a transform species

(Pysek et al., 2004) because it leads to a community transformation from grassland to woodland (Bruce et al., 1995; Battaglia et al., 2009); what is worse, it also results in the huge economic loss because of the loss of timber production, and extra control cost

(Wang et al., 2012). Now, tallow is rapidly spreading and expanding into previously un- 1

infested regions include non-coastal areas and further inland regions (Pattison and Mack,

2009; Wang et al., 2011; Fan et al., 2012). Taking Mississippi as an example, Oswalt

(2010) reported the population of Chinese tallow in Mississippi increased 445 percent from 1994 to 2006 based on data collected by the U.S. Department of Agriculture Forest

Service, Southern Research Station, Forest Inventory and Analysis Program (FIA).

Additionally, tallow grows faster in introduced regions than in native areas (Zou et al., 2008). At three years old, it can produce seeds and become mature which is the reason that tallow is such a severe threat to the southern forestland. Growth models are of great importance in projecting future forest invasion conditions and they provide an accurate method for resource managers to predict the growing stock of tallow in forests for estimating native timber loss. Moreover, information provided by growth models for tallow tree is helpful in the management of infested forestlands and designing policy strategies. Therefore, the specific objectives of this study are as follows: 1) develop growth models that include the variables DBH/age, height/age, volume/age, and biomass/age, as well as the allometric relationships between height/DBH and volume/biomass with height and DBH on two coastal flatland and bottom-and-low land ecosystems dominated by oak-gum-cypress (Quercus/Liquidambar styraciflua/Taxodium distichum) and longleaf/slash pine forest (Pinus taeda/ Pinus echinata); 2) build tree

profile equations for tallow and explore if stem taper is influenced by site conditions.

CHAPTER II

LITERATURE REVIEW

2.1 Studies on exotic tallow tree

Chinese tallow tree has posed a sever threat to native ecosystems because it aggressively displaced native plants even formed mono-specific forests (Bruce et al.,

1997). Hence, studies on what factors contribute to the successful invasion of tallow appeared and most of them mainly focused on the difference of characteristics of tallow between invasion region and native areas (Zou et al., 2006, 2007, 2008; Gan et al., 2009).

For instance, studies (Scheld and Cowles, 1981; Canmeron and Spencer, 1989; Conner,

1994) show that tallow could tolerate a set of soils such as sandy, clay, and saline; the productive waxy seeds (approximately100, 000 per year) could be dispersed by birds, water, and human (Jin and Huang, 1984; Bruce et al., 1997; Renne et al., 2002); seedlings grew rapidly (Zou et al., 2008), tolerated of light (Rogers and Siemann, 2002) and nutrient conditions, and had highly competitiveness ability (Siemann and Rogers, 2003;

Pattison and Mack, 2008; Zou et al., 2008); all of these characters facilitated the fast infestation of tallow in southern forestlands. Moreover, disturbance which changed the resource availability had a critical contribution to successful invasion of tallow (Burke and Grime, 1996). Gan et al (2009) reported that disturbances like timber harvesting, damage caused by animals, wind, and fire all contributed to tallow invasion. Furthermore,

private forestlands were more severely invaded than public ones because there were more mature trees and less frequent harvests in public forests (Gan et al., 2009).

Based on these contributing characteristics and factors, modeling the relationship between the rapid spread of tallow and associated driving factors inspired a new wave of research on tallow. Gan et al. (2009) and Wang et al. (2011) applied logistic regression in connecting the probability of tallow occupation with a series of factors including site condition, geographic characters, and disturbances. However, when spatial autocorrelation of invasive species caused by dramatic change of driving factors over space was considered, non-spatial logistic regression was not accurate (Fan et al., 2012).

Consequently, Fan et al. (2012) presented a modified approach to quality the driving factor effect which was composed of two steps: classification and regression tree and logistic regression. Taking spatial dependence into consideration, auto-logistic regression and simultaneous autoregressive regression (SAR) methods were also used to link the impact of driving factors and spread of invasion species (Tan, 2012). Modeling the driving factors and tallow presence provides a theoretical basis for predicting potential invasion regions. Preceding research reported that the probability of tallow trees migrating farther north increased with the rise of temperature (Pattison and Mack, 2008;

Gan et al., 2009; Wang et al., 2011). Pattison and Mack (2008) used the CLIMAX model to predict that tallow would extend 700 km northward; Wang et al. (2011) estimated that tallow would move over 300 km to the north (as north as 340N).

In addition, there are also numerous studies on designing efficient measures to eradicate tallow tree from forests (Jubinsky et al., 1996; Bruce et al., 1997; Barrilleaux et al., 2000; Kolar et al., 2001; Grace et al., 2001; Burns et al., 2004), but all these measures

are costly and are not available to some areas. Wang et al. (2012) presented that complete

eradication was not cost-effective and suggested early detection and control was the key

to managing Chinese tallow. Currently, it is still a regional concern to design efficient

and effective measures to mitigate and control further invasion of tallow.

2.2 Growth models of tree species

Tree growth can be defined (Avery and Burkhart, 1983) as the increment in

diameter, height, volume, or biomass over a given period. Traditionally, there are three

categories of empirical growth models: whole stand models, diameter class models, and

individual tree models. Stand models are established on the basis of stand level

information and predicted the growth of the whole stand (Ochi and Cao, 2003;

Huuskonen et al., 2007), while diameter class models predicted the growth rates of stand

by diameter class (Solomon et al., 1995; Eerikäinen et al., 2003). Individual tree growth

models are developed for predicting the growth of each individual tree in a certain stand

(e.g. Hynynen et al., 2002; Zhao et al., 2003; Sironen, 2009). Individual tree growth

models are further divided into distance-independent and distance-dependent models

which take spatial correlation into consideration (Porté et al., 2002; Aguirre et al., 2003;

Zhang et al., 2004); however, due to the complexity and high cost of mapping each tree

location, distance-dependent models are not as widely used as distance-independent

models (Munro, 1974; Wimberly and Bare, 1996; Porté et al., 2010).

Growth models, in general, are built based on two methods: physiological mechanism of trees (process models) and empirical methods (Vanclay, 1994). However, due to the constraints of process models which are influenced by complicated environment factors, empirical models are more commonly used in practice to predict 5

future forest situations (Vanclay, 1994). Diameter at breast height (DBH) is the most used and easiest measurement of tree size in growth and yield modeling (Avery and

Burkhart, 1983). Height is another traditionally measured tree attribute used to estimate the volume of an individual tree or forest (Hann et al., 1991). Basal area equations based on diameter are the most linearly related to volume growth of all those developed (Hokka et al., 1999). The relationship between volume and diameter and height is the most common procedure to estimate tree volume (LeMay, 2006). For example, Bi and

Hamilton (1998) reported that Spurr’s volume equation within which volume was linear to diameter square times height was commonly used; later, the Schamacher volume equation within which volume was nonlinear to the diameter and height appeared (Laar and Akca, 2007).

For modeling the increment of individual tree diameter or basal area, a composite or a modifier model as developed by Zhang et al. (2004). In a composite model, independent variables usually refer to tree characteristics (such as tree size, crown ratio, vigor) and stand conditions (such as stand age, site index, and stand density); while the responsible variables are either diameter or basal area increment (e.g., Vanclay, 1994;

Zhang et al., 2004). By contrast, a modifier model represents a potential maximum attainable growth for a tree, and it was able to explain tree growth biologically (Zhang et al., 1997). In recent studies (Zhang et al., 2004; Budhathoki et al., 2008) potential growth was also modeled using more flexible functions such as the Chapman-Richards and logistic models which are better explained ecologically. However, if non-homogeneity of variance exists during construction of tree volume growth model, it is difficult to estimate the parameters with the least squares method. Therefore, a modified modeling approach

using weighted least square regression was developed. Meng and Tsai (1986) built a volume growth model for loblolly pine and compared the weight values of diameter- squared*height (D2H) and diameter-squared (D2) in the model. Diéguez et al. (2006) developed an individual tree volume model for Scots pine (Pinus sylvestris L.) using a modified second-order continuous autoregressive method mainly to resolve the high autocorrelation of the data.

2.3 Tree profile functions

Taper is defined as the rate of change in diameter with height, which varies with species, age, DBH, and stand condition (Husch et al., 1982); therefore, taper describes diameter changes with height along the stem (Methol, 2001; Ounekham, 2009). With tree profile equations, people can predict diameters at any height along the bole of a tree, height for a given diameter, and individual log volumes (Methol, 2001). Taper equations are the mathematical function of the diameter change with respect to height on the basis of species, age, and stand condition (Husch et al., 1982; Brooks et al., 2008). In the past

100 years, numerous taper functions have been developed for different tree species with various forms from simple ones (Kozak et al., 1969; Ormerod, 1973; Hilt, 1980;

Zakrzewski et al., 2006) to complex (Max and Burkhart, 1976; Cao et al., 1980; Clark et al., 1991; Jiang et al., 2005). Methol (2001) classified taper equations into four categories: single functions, within-tree variable form, between-tree variable form, and segmented polynomial models; while Jiang et al. (2005) summarized them into three classifications: simple taper functions, variable form taper functions, and segmented polynomial taper functions.

Simple taper functions mainly define tree profiles with a single continuous equation for the whole bole (Bruce et al., 1968; Hilt, 1980; Gordon et al., 1995).

However, these simple taper equations were unable to precisely describe the whole bole profile, although they could reflect the general stem form (Jiang et al., 2005). Therefore, more complicated and accurate taper functions appeared. Max and Burkhart (1976) developed a segmented polynomial regression method to build a profile equation for loblolly pine (Pinus taeda) evaluated by Clark et al. (1991) as performing well in predicting diameter. Sharma and Burkhart (2003) described it as the combination of three sub-models at two join points. However, the application of Max and Burkhart model is limited to a small range of diameters and heights due to its complexity (Matney and

Parker, 1992; Parker, 1997). Cao (2009) modified the Max and Burkhart loblolly pine model by calibrating DBH and upper stem diameter. Later, Clark et al. (1991) developed a form-class segmented profile model which provided volume estimation more accurately than the segmented Max and Burkhart model. Souter (2003) employed a segmented profile function in southern tree species to predict diameter at a specified height. In summary, segmented polynomial functions, composed of a series of sub-models representing various sections of the stem better than simple and variable form taper functions and are widely used (Ounekham, 2009).

CHAPTER III

METHODS

3.1 Data Collection

A total of 33 sample trees were selected in this study and 11 of them were felled in an oak/gum/cypress bottomland forest in southern Mississippi (Old River Wildlife

Management Area in Poplarville city, Pearl River County, MS); 16 were sampled from the Grand Bay National Estuarine Research Reserve (NERR) (Jackson County, MS) and an additional 6 trees were obtained from the Mississippi Sandhill Crane National Wildlife

Refuge (NWR) (Jackson County, MS) in the Gulf Coast Complex. The Grand Bay and

Sandhill are primarily longleaf/slash pine forest. Sample tallow trees were chosen to ensure a representative distribution of both the diameter and height on the corresponding sites. To obtain the sample trees age, destructive sampling was used and the details of the process is shown in Figure 3.1. Before felling sample trees in the field, total height and diameter at breast height (DBH) ware measured. Trees were all felled at ground level for stem analysis and then the stem was divided into one-meter sections. Disks with 3-5 centimeters thickness were then extracted from the mid-point of each section (Figure

3.2). Each disk was marked with location name, tree number, and disk number

(representing the stem position where it was extracted). Next, all disks were placed into a plastic bag to maintain the fresh weights. The diameter outside and inside bark was

measured at the lower and upper end of each section. Diameter at selected height positions (0.84 and 5.3 m) was also recorded.

After all these disks were returned from the field, fresh weight was first measured on each with a digital scale and then the diameter outside and inside bark was also measured. To estimate stem biomass accurately, each disk was divided into bark and wood. Then volumes of each disk in cubic meters were obtained by immersing it into a

(15 x 30 x 35 cm) container with water to measure the volume of water displaced (based on Archimedes Principle). Next, age was obtained by counting the visible rings on each sanded disk. Finally all disks were dried in an oven for 24 hours (at 120°C) to measure dry weight. This work was completed in the U.S. Forest Service Wood Products Insect

Research Lab in Starkville, MS. Volumes and biomass of each section were calculated using equations (3.1) and (3.2), respectively.

� v = (�2 + �2) ∗ l (3.1) 8

Where: � is the length of each section (1 meter); � and � are diameters at upper and lower end of each section. The whole stem volume of individual tree is acquired (��) by adding up all section volumes. Then the dry weight of each section was computed with

Equation 3.2:

�� = (3.2) ��

Where: �� is the dry weight of each one-meter section; �� indicates fresh volume of each one-meter sections (�); �� represents the dry weight of disks weighed by digital scale and �� is fresh volume of disks which is measured with water displacement method in a container. Finally, the total dry weight of individual tree wood can be computed by 10

adding up ��for all one-meter sections (Fig. 3.1). All variables measured or calculated were summarized in Table 3.1.

Figure 3.1 Flow chart outlying experiment methods in the field and laboratory

Figure 3.2 Disk abstraction (circle) and section separation (slash) along tree stem

Table 3.1 Descriptive statistics for sample tallow trees in Poplarville, Grand Bay, and Sandhill, MS

Sites Statistics DBH (cm) Total Height (m) Volume (m3) Biomass(Kg)

Mean 17.8 13.2 0.140 59.6 Poplaville Min 7.9 7.4 0.014 6.0 (11 trees) Max 29.9 19.0 0.430 186.6

Mean 11.4 10.7 0.053 28.8 Grand Bay Min 4.0 7.0 0.004 1.8 (16 trees) Max 34.5 18.8 0.300 161.6

Mean 13.9 13.0 0.054 27.6 Sand Hill Min 6.7 9.4 0.012 6.0 (7 trees) Max 25.5 21.0 0.110 52.8

Figure 3.3 Boxplot for DBH, height, volume, and biomass collected from Poplarville (Poplar), Grand Bay (Grandbay), and Sandhill sites

3.2 Model Development

3.2.1 Height, Diameter, and Volume/Biomass equations

To fit growth models for height/age, DBH/age, and the allometric relationship between DBH and height, the Schumacher, Chapman-Richards, logistic, and Mitscherlich models were selected (see Equations 3.3, 3.4, 3.5, and 3.6). 13

� − � = � ∗ � � (3.3)

� = �(1 − �−��)� (3.4)

� � = �, �, � > 0 (3.5) �+��−��

� = �(1 − �−��) �, � > 0 (3.6)

In all these models �, �, �� are the parameters to be estimated; � is the cumulative

growth of diameter or height in the measurement year and � indicates the age of tallow.

These four models were fitted using SAS (SAS Version 9.2, SAS Inc., Campus Drive, Cary,

NC 27513) and then, the best data fitting model was selected. Afterwards, corresponding model form and model parameters was compared to test the difference of growth rate of tallow at three sample sites.

There are many different linear or nonlinear functions commonly used for tree volume growth models. Generally, stem volume is a function of diameter, height, and stem form. However, it is more difficult and expensive to obtain a stem profile function under most circumstances; therefore, the allometric relationship among volume and diameter and height was only established here. The most common linear equation form for volume is Spurr’s volume equation (Bi and Hamilton, 1998), and the mathematical expression is given in Equation 3.7 where �0, �1 are the model parameters to be estimated.

2 �� = �0 + �1�� × ℎ��ℎ� (3.7)

A stem biomass accumulation model was also constructed based on volume models. Accordance with the conception of biomass, an individual tree is composed of above-ground and below-ground portions. Above-ground biomass refers to stem, stump,

branches, bark, seeds, and foliage while below-ground biomass consists of all living roots

excluding fine roots (less than 2mm in diameter) (Samalca, 2007). Nevertheless, due to

the data limitation, only stem biomass (dry weight) was established in this study. The linear model development process was similar to that of the volume model.

3.2.2 Taper Functions

According to Ounekham (2009), segmented polynomial models were commonly used in fitting tree profile equations because of their accuracy and good performance.

Therefore, the segmented polynomial approaches of Max and Burkhart (1976), Cao

(2009), and Clark, Souter, and Schlaegel (1991) were selected to fit the stem profile of

tallow in this study.

The common notations for the taper equations are listed below and will be used

hereafter.

�: The total height (m);

ℎ�: Breast height (1.3 m);

�: Diameter at breast height (DBH) (cm);

�: Diameter at height ℎ (cm);

ℎ: Height above the ground to the measurement point (m);

The well-known Max and Burkhart profile function described the bole of a tree with three sectors according to generalized relative geometric form (Clark, et al., 1991).

The form of segmented polynomial taper equation (Equation 3.8) is:

2 2 0.5 � ℎ ℎ2 ℎ ℎ = {(�1 ( − 1) + �2 ( − 1) + �3 (�1 − ) �1 + �4 (�2 − ) �2)} (3.8) � � �2 � �

ℎ The constraints of this model are: ℎ = �, � = 0, �� ℎ = ℎ�, � = �. �1 = 1 if ≤ �1, �

ℎ and�1 = 0, otherwise; �2 = 1 if ≤ �2, and �2 = 0, otherwise. �1, �2, �3, �4, �1, �2 are �

parameters to be estimated.

Cao (2009) reported that the Max and Burkhart (1976) taper model was inaccurate

in predicting upper-stem diameter; and modified the model as given in Equations (3.9)

and (3.10).

2 2 ∗ ℎ ∗ ℎ ℎ ℎ � (1 − ) = � × (1 − ) + �2 (1 − ) + �3 × �1 (1 − − �1) + �4 × � 1 � � �

ℎ 2 �2 (1 − − �2) (3.9) �

1.37 1.37 1.37 �(1− )−�̂(1− )+� ×(1− ) ∗ � � 1 � �1 = 1.37 (3.10) (1− ) �

∗ ∗2 2 ∗ Where: � = � ⁄� and � is calibrated diameter; the parameter �1is modified to

∗ parameter �1. The remaining variables retained their same meaning as in the Max and

Burkhart model.

A form-class segmented taper model was developed by Clark et al. (1991) composing of four different sections: butt section (from stump to 1.37 m), lower stem

(from 1.37 m to 5.3 m), middle stem (from 5.3 m to 40–70% of total height), and upper stem (from 40% to 70% of total height to the tip of the tree) (Jiang et al. 2005; Özçelik et al. 2012) as given in Equation (3.11).

�1 � � �4 �3 ℎ 1.37 1 2 2 1.37 4 ℎ (�2+ 3)((1− ) −(1− ) ) (� −� )((1− ) −(1− ) ) 2 � � � 2 � � � = {�� {� (1 + )} + �� {� − } + 1.3 �1 1.37 �4 5.3 �4 (1−(1− )) ((1− ) −(1− )) � � �

2 2 2 ℎ−5.3 1−�6 ℎ−5.3 0.5 ��{� (�6 ( − 1) + �� ( 2 ) (�5 − ) )}} (3.11) �−5.3 �5 �−5.3

Where: � is the diameter at 5.3 m, �1, �2, �3is the regression coefficient for the butt section, �4 is the coefficient for lower stem, and �5, �6 are the coefficients for height

above 5.3 m. Hence, the four constraints in Equation (3.11) are defined as following:

1 ℎ < 1.37 � = { � 0 ��ℎ�� 1 1.37 < ℎ < 5.3 � = { � 0 ��ℎ�� 1 ℎ > 5.3 � = { � 0 ��ℎ��

1 ℎ < (5.3 + � (� − 5.3)) � = { 5 � 0 ��ℎ��

3.3 Models Evaluation

Final best fitted model selection was based on a series of statistical interferences and the capability of the model to reflect the ecologic growth process of tallow. R square

(�2) (Kvålseth, 1985), bias, fit index (FI), and root mean square error (RMSE) were used

to evaluate the model fitting degree (Schlaegel et al., 1981). Their mathematical

definitions as follows:

� ̂ 2 ∑ �=1(� � − ��) � − � − 1 �2 = 1 − � ̅ 2 ∑ �=1(� � − �) � − 1

∑� (� −�̂ ) �� = �=1 � � �

� ̂ 2 ∑ �=1(� �−��) �� = 1 − � ̅ 2 ∑ �=1(� �−�)

�� = √�̅2 + ��2

� ̂ �̅ = ∑ (�� − ��)⁄� �=1

Where: SD is standard deviation of the predicted errors, �� is the prediction error or

difference between observations (��) and the predictions (�̂�); �̅ is the mean of ��. � is the

number of observations in the data set while � represents the number of estimated

parameters in the model. �2 was assessed for diameter, height, volume, and biomass growth models and the allometric relationship models among volume, and biomass and diameter and height; while for taper models, the model performance is evaluated by bias, the root mean squared error (RMSE) and a fit index (FI) (Schlaegel et al., 1981).

Additionally, graphical residual plotting against predicted values and independent variables, were also used in this study.

CHAPTER IV

RESULTS

4.1 Sites effects test

Sample tallow trees were chosen from three different sites and different forest components and soil condition within each site. The sample plot in Poplarville is a southern oak/gum/cypress forest while Grand Bay NERR and Sandhill Crane NWR, located in the extreme southeast Jackson County, MS. are dominated by longleaf/slash pine. According to the summary descriptive statistics of DBH, height, volume, and biomass observations in these three stands (Table 3.1), the attributes of tallow varied with location. Figs 4.1 and 4.2 show the trends of growth models and taper on three sites based on observations. To describe this difference statistically, differences among the sites were tested using SAS before quantifying growth models and taper functions for all sample tallow trees on these three locations. Sandhill was designed as a reference site in the SAS test. Based on testing results (Table 4.1): (1) for DBH/age, height/DBH, volume/age, and biomass/age growth models and taper equation, the observations at Poplarville oak/gum/cypress forest were significantly different (P-value < 0.05) from the other two sites; (2) for height/age and volume/DBH models, there was no significant difference among them which indicated that these two models were the same for all three sites; (3) for allometric relationship between volume/biomass and DBH square times height, Grand

Bay (P-value < 0.001) was significantly different from the other two sites. Therefore, 19

models including DBH/age, height/DBH, volume/age, and biomass/age and the profile function needed to be quantified uniquely for tallow at the Poplarville site; and for height/age and volume/DBH, we could describe them using the same models for all three sites. By contrast, the relationship model among volume/biomass and DBH square times height had to be built separately for the Grand Bay site.

Table 4.1 SAS results for DBH, height, volume, biomass growth models and profile functions for tallow tree at Poplarville, Grand Bay, and Sandhill (the reference point), MS, United States.

Models Sites P-value

DBH/age Poplarville <0.0001 Grand Bay 0.5571 Height/age Poplarville 0.2240 Grand Bay 0.5584 Height/DBH Poplarville 0.0154 Grand Bay 0.2092 Volume/age Poplarville 0.0006 Grand Bay 0.5224 Volume/DBH Poplarville 0.6765 Grand Bay 0.2094 Volume/DBH2*Height Poplarville 0.1680 Grand Bay 0.0027 Biomass/age Poplarville 0.0084 Grand Bay 0.5801 Biomass/ DBH2*Height Poplarville 0.4190 Grand Bay <0.0001 Taper Poplarville <0.0001 Grand Bay 0.1348

Figure 4.1 Height/DBH, DBH/age, height/age, and volume/age trends based on observations of tallow at Poplarville (blue dots), Grand Bay NERR (green diamonds), and Sandhill NWR (red stars)

Figure 4.2 Biomass/age, volume/DBH, and volume/biomass/DBH2*height trends based on observations of tallow at Poplarville (blue dots), Grand Bay NERR (green diamonds), and Sandhill NWR (red stars)

Figure 4.3 DOB (A and C) and DIB (B and D) stem profile trends based on measurements of tallow at Poplarville, Grand Bay, and Sandhill, MS, sample sites. 23

Figure 4.3 (continued)

4.2 Fitted Models

4.2.1 Diameter, Height, and Volume equations

All Equations 3.3, 3.4, 3.5, and 3.6 were fitted for height and DBH cumulative growth models using SAS. Then, based on R-square the best fitted models of DBH/age,

height/age, height/DBH for tallow at Poplarville, Grand Bay, and Sandhill were selected

and summarized in Table 4.2. It can be concluded from the models that: (1) DBH of

tallow grew faster (� = 40.30414, � = −7.66194) in the Poplarville oak/gum/cypress

forest than that at Grand Bay and Sandhill longleaf/slash pine forest (� = 34.5441, � =

−8.695), although they both grew exponentially with age; (2) by contrast, height/age

(� = 3.1923, � = 0.6) and volume/DBH (� = 6.86, � = 2.3317) models of tallow

were in power functions for all three sample stands; (3) for the height/DBH model, the

height of tallow at Poplarville Oak/Gum/Cypress forest changed with DBH in an

exponential way (� = 22.01, � = −8.534) while tallow at Grand Bay and Sandhill

longleaf/slash pine forest showed a power trend (� = 3.22, � = 0.522); moreover, tallow 24

was taller in the longleaf/slash pine forest at Grand Bay and Sandhill than in the oak/gum/cypress forest at Poplarville with the same DBH; (4) volume and biomass growth models showed that both volume (� = 0.9375, � = −19.4687) and biomass

(� = 207.1344, � = −19.56) grew faster in oak/gum/cypress forest at Poplarville than

trees at Grand Bay and Sandhill longleaf/slash pine forests; and (5) the allometric

relationship between volume/biomass and DBH square times height model showed that

volume (�� = 0.294, �� ℎ�� = 0.254) and biomass

(�� = 81.51, �� ℎ�� = 54.172) of tallow at Grand Bay grew

faster than that at Poplarville and Sandhill with DBH and height based on the slope �.

Table 4.2 Parameters estimated for DBH, height, and volume/biomass growth models and allometric relationship models among them

Models Sites Model Form R2

1 Poplarville −7.66194× �� = 40.30414 × � �� 0.7722 DBH/age

Grand Bay 1 −8.695×�� 0.9116 Sandhill �� = 34.5441 × �

Poplarville

Height/age Grand Bay ��ℎ� = 3.1923 × ��0.6 0.8567 Sandhill

1 Poplarville −8.534× ��ℎ� = 22.01 × � �� 0.7511 Height/DBH Grand Bay ��ℎ� = 3.22 × ��0.522 0.9369 Sandhill

1 −19.4687×�� Poplarville �� = 0.9375 × � 0.7647 1 Volume/Biomass −19.33× 0.7854 �� = 404.96 × � �� /age Grand Bay �� = 0.00013 × ��2.71 0.9487 Sandhill �� = 0.043 × ��2.89 0.9438 Poplarvile

Volume/DBH Grand Bay �� = 6.86 × ��2.3317 0.989 Sandhill

�� = 0.0075 + 0.254 × ��2 Poplarville 0.9893 × ℎ��ℎ�

Sandhill �� = 5.39 + 108.33 × ��2 0.9865 Volume/Biomass × ��ℎ� 2 /DBH ×Height �� = 0.0017 + 0.294 × ��2 Grand Bay 0.9975 × ℎ��ℎ�

�� = 0.13 + 163.1 × ��2 0.9953 × ℎ��ℎ�

4.2.2 Taper Functions

4.2.2.1 Model Parameters

Max and Burkhart (1976), the Cao (2009) modification of Max and Burkhart, and

Clark et al. (1991) models were all fitted in T-Profile software (Matney, 1996). Both

DOB and DIB at different heights were measured with the destructive sampling method; hence, all three models were fitted for both diameter-inside-and-outside barks. For the inside bark taper fitting process, DIB replaced DOB with the other variables being the same as with the DOB fitting process. Tables 4.3, 4.4, and 4.5 showed the parameter

estimation for these three different profile models at sample regions. Figs 4.3 and 4.4

showed the fitted stem profile based on the three segmented profile models for tallow at

Poplarville oak/gum/cypress forest and Grand Bay and Sandhill longleaf/slash pine

forests.

Table 4.3 Estimated parameters and fit statistics for the Max and Burkhart profile model for tallow at Poplarville, Grand Bay, and Sandhill, MS, sites.

Sites Poplarville Grand Bay and Sandhill Parameter OB profile IB profile OB profile IB profile a -1.04 -26.57 -19.69 -18.88 b -0.02 12.90 9.10 8.72 c 42.49 32.67 9.38 9.33 d 0.76 -12.93 -9.90 -9.62 alpha 0.077 0.095 0.225 0.222 beta 0.350 0.990 0.987 0.987 No. Obs. 281 281 410 410 RMSE 0.060 0.057 0.088 0.088 Index of Fit 0.964 0.962 0.912 0.876

Table 4.4 Estimated parameters and fit statistics of the Cao (2009) profile model for tallow at Poplarville, Grand Bay and Sandhill, MS, sites.

Sites Poplarville Grand Bay and Sandhill Parameter OB profile IB profile OB profile IB profile a 1.88 -42.19 -19.76 -18.95 b -1.49 20.88 9.13 8.75 c 28.2 29.43 9.16 9.20 d 1.54 -20.94 -9.98 -9.68 alpha 0.10 0.10 0.230 0.230 beta 0.98 0.99 0.986 0.987 No. Obs. 270 270 389 389 RMSE 0.060 0.061 0.090 0.088 Index of Fit 0.964 0.956 0.910 0.883

Table 4.5 Estimated parameters and fit statistics of the Clark, Souter and Schlaegel profile model for tallow at Poplarville, Grand Bay and Sandhill, MS, sites.

Sites Poplarville Grand Bay and Sandhill Parameter OB profile IB profile OB profile IB profile a 20.98 22.24 17.83 18.72 b 0.946 0.998 1.11 1.19 c 84.17 65.00 84.33 57.93 d 2.45 2.56 2.34 2.18 e 1.01 1.07 2.19 2.47 alpha 0.50 0.15 0.798 0.799 No. Obs. 259 259 376 0.799 RMSE 0.052 0.057 0.069 0.078 Index of Fit 0.974 0.968 0.947 0.930

Figure 4.4 Fitted DOB stems profiles for tallow tree at Poplarville MS (A) and Grand Bay/Sandhill MS (B) sites using Max and Burkhart (dash), Cao (2009) (round dot), Calrk et al. (1991) (solid)

Figure 4.5 Fitted DIB stem profiles for tallow tree at Poplarville MS (C) and Grand Bay/Sandhill MS (D) sites using Max and Burkhart (dash), Cao (2009) (round dot), Calrk et al. (1991) (solid)

4.2.2.2 Comparison of Taper Models

Three different segmented polynomial taper models were fitted for exotic tallow

tree at three different locations in southern Mississippi. Kozak and Smith (1993) reported that when choosing a best stem taper model for application, people needed to consider both practical and statistical criteria. As a consequence, the standards to choose the best fitted taper model for tallow in these locations were based on both statistical variables 30

(RMSE and FI) and the capability of the model to describe the ecological growth process.

RMSE and FI are presented in Tables 4.6 and 4.7. First, the fitted results show that: for

DOB profile of tallow in an oak/gum/cypress forest at Poplarville, MS, all three models performed well; although based on RMSE and FI, Clark et al. (1991) (RMSE=0.052,

FI=0.974) was better than the Max and Burkhart taper model (RMSE=0.060, FI=0.964) and Cao (2009) modified Max and Burkhart model (RMSE=0.060, FI=0.964), but there was no significant difference among them. Likewise, for DOB of tallow in

Longleaf/Slash pine forest at Grand Bay and Sandhill, these three profile models also performed with no apparent difference though Clark et al. (1991) performed better

(RMSE=0.069, FI=0.947) than Max and Burkhart (RMSE=0.088, FI=0.912) and the Cao

(2009) modification model (RMSE=0.090, FI=0.910) on the basis of RMSE and FI;

Second, for the DIB profile of tallow at Poplarville, there was also no evident difference

among the three fitted models: Max and Burkhart (RMSE=0.057, FI=0.962), Cao (2009)

modification model (RMSE=0.061, FI=0.956), and Clark et al. (1991) (RMSE=0.057,

FI=0.968). However, for DIB profile of tallow at Longleaf/Slash pine forest, Clark et al.

(1991) (RMSE=0.078, FI=0.930) performed better based on RMSE and FI while the

other two models exhibited the same performance (RMSE=0.088, FI=0.880).

Table 4.6 RMSE and FI for the estimated diameter outside bark (DOB) profiles of tallow at Poplarville and Grand Bay/Sandhill, MS, sites

DOB profiles Profile Model Poplarville Grand Bay and Sand Hill Max and Burkhart RMSE 0.060 0.088 Index of Fit 0.964 0.912 Clark, Souter and RMSE 0.052 0.069 Schlaegel Index of Fit 0.974 0.947

Cao (2009) RMSE 0.060 0.090 Index of Fit 0.964 0.910

Table 4.7 RMSE and FI for the estimated diameter inside bark (DIB) profiles of tallow at Poplarville and Grand Bay/Sandhill, MS, sites

DIB profiles Profile Model Poplarville Grand Bay and Sand Hill Max and Burkhart RMSE 0.057 0.088 Index of Fit 0.962 0.880 Clark, Souter and RMSE 0.057 0.078 Schlaegel Index of Fit 0.968 0.930

Cao (2009) RMSE 0.061 0.088 Index of Fit 0.956 0.880

To compare the three different taper models for tallow at Poplarville and Grand

Bay and Sandhill, details of RMSE, biases, and residuals for both DOB and DIB are given in Tables 4.8 to 4.17. Residual plots are also shown in Figs 4.5 and 4.6. Residual

plot 4.5 showed that: Clark et al. (1991) model for DOB profile of tallow at Poplarville

was the least biased model (-0.03

and Burkhart model (-0.04

the same bias range. Similarly, Clark et al. (1991) was also the least biased model for

DOB of tallow at Grand Bay and Sandhill (-0.03

Burkhart model at the upper section of stem. Likewise, Fig. 4.6 displays the residuals of these three models for DIB of tallow at Poplaville versus Grand Bay and Sandhill and revealed a similar trend. The residual ranges of the three models for DIB profile of tallow at Poplaville were: Max and Burkhart (-0.04

0.04

Grand Bay and Sandhill the range were: Max and Burkhart and Cao (2009) (-

0.06

Table 4.8 Residuals, RMSE, and bias of Max and Burkhart taper model for DOB at Poplarville, MS.

Relative height n Residual RMSE Bias 0.00(0.01) 12 -0.01 0.10 -0.01 0.05(0.05) 18 0.01 0.06 0.00 0.10(0.10) 16 0.00 0.04 0.00 0.15(0.15) 16 -0.01 0.03 -0.01 0.20(0.20) 14 0.01 0.04 0.01 0.25(0.25) 18 0.00 0.05 -0.01 0.30(0.30) 18 -0.01 0.06 -0.02 0.35(0.35) 17 0.00 0.05 0.00 0.40(0.40) 20 -0.01 0.04 -0.02 0.45(0.45) 19 0.00 0.04 0.00 0.50(0.50) 14 0.01 0.06 0.02 0.55(0.55) 18 0.00 0.06 -0.02 0.60(0.60) 16 0.01 0.08 -0.03 0.65(0.65) 16 -0.01 0.07 -0.07 0.70(0.70) 10 0.02 0.08 0.01 0.75(0.75) 11 -0.01 0.08 -0.13 0.80(0.79) 8 0.01 0.09 -0.13 0.85(0.87) 9 -0.04 0.09 -0.76 0.90(0.93) 8 0.03 0.09 0.02 0.95(0.97) 12 -0.01 0.02 -0.50 1.00(0.99) 2 0.00 0.00 -0.04

Table 4.9 Residuals, RMSE, and bias of Cao (2009) model for DOB at Poplarville, MS.

h/H n Residual RMSE Bias 0.00(0.01) 12 -0.03 0.11 -0.03 0.05(0.06) 24 -0.01 0.04 -0.01 0.10(0.11) 11 0.00 0.02 0.00 0.15(0.15) 16 -0.01 0.03 -0.01 0.20(0.20) 13 0.00 0.03 0.00 0.25(0.25) 18 -0.02 0.06 -0.03 0.30(0.30) 18 -0.03 0.06 -0.04 0.35(0.35) 17 -0.01 0.05 -0.02 0.40(0.40) 20 -0.02 0.05 -0.04 0.45(0.45) 19 -0.01 0.04 -0.02 0.50(0.50) 14 0.00 0.05 0.00 0.55(0.55) 18 -0.01 0.06 -0.03 0.60(0.60) 16 0.00 0.07 -0.03 0.65(0.65) 16 -0.01 0.07 -0.07 0.70(0.70) 10 0.02 0.08 0.00 0.75(0.75) 11 -0.01 0.07 -0.12 0.80(0.79) 8 0.01 0.09 -0.11 0.85(0.87) 9 -0.04 0.09 -0.74 0.90(0.93) 8 0.03 0.09 0.03 0.95(0.97) 12 -0.01 0.02 -0.50 1.00(0.99) 2 0.00 0.00 -0.02

Table 4.10 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DOB at Poplarville, MS.

h/H n Residual RMSE Bias 0.00(0.01) 12 -0.01 0.10 -0.01 0.05(0.06) 24 0.00 0.04 0.00 0.10(0.11) 11 0.01 0.03 0.01 0.15(0.15) 16 0.00 0.03 0.00 0.20(0.20) 13 0.01 0.03 0.01 0.25(0.25) 19 -0.01 0.05 -0.01 0.30(0.30) 17 -0.01 0.04 -0.02 0.35(0.36) 20 0.00 0.03 0.00 0.40(0.41) 19 0.00 0.02 0.00 0.45(0.45) 17 0.01 0.03 0.01 0.50(0.50) 14 0.02 0.04 0.03 0.55(0.55) 18 0.00 0.05 0.00 0.60(0.61) 17 0.01 0.06 -0.01 0.65(0.65) 16 0.00 0.06 -0.05 0.70(0.70) 9 0.01 0.08 -0.02 0.75(0.75) 11 -0.01 0.07 -0.10 0.80(0.79) 8 0.00 0.08 -0.15 0.85(0.87) 9 -0.04 0.08 -0.70 0.90(0.93) 8 0.03 0.08 0.00 0.95(0.97) 12 -0.01 0.02 -0.46 1.00(0.99) 2 0.00 0.00 -0.06

Table 4.11 Residuals, RMSE, and bias of Max and Burkhart taper model for DOB at GrandHill, MS.

h/H n Residual RMSE Bias 0.00 (0.02) 21 0.06 0.13 0.04 0.05 (0.05) 27 -0.06 0.12 -0.06 0.10 (0.10) 22 -0.01 0.06 -0.01 0.15 (0.15) 24 0.02 0.05 0.02 0.20 (0.20) 23 0.04 0.05 0.04 0.25 (0.25) 24 0.02 0.05 0.02 0.30 (0.30) 26 -0.01 0.04 -0.01 0.35 (0.35) 23 -0.02 0.06 -0.03 0.40(0.40) 22 -0.01 0.05 -0.02 0.45 (0.45) 26 -0.02 0.06 -0.03 0.50 (0.50) 24 -0.03 0.08 -0.05 0.55 (0.55) 23 -0.04 0.08 -0.09 0.6 0(0.60) 19 -0.02 0.09 -0.07 0.65 (0.65) 21 -0.01 0.06 -0.03 0.7 0(0.70) 17 0.01 0.08 -0.02 0.75 (0.74) 16 0.04 0.09 0.06 0.80 (0.80) 17 0.05 0.09 0.11 0.85 (0.89) 14 0.08 0.12 0.20 0.9 0(0.95) 31 0.08 0.12 0.24 0.95 (0.97) 6 0.03 0.05 0.21 1.00 (1.12) 5 0.18 0.33 0.44

Table 4.12 Residuals, RMSE, and bias of Cao (2009) profile model for tallow DOB at GrandHill, MS.

h/H n Residual RMSE Bias 0.00 (0.02) 21 0.04 0.10 0.02 0.05 (0.05) 33 -0.05 0.08 -0.05 0.10 (0.10) 28 0.00 0.03 0.00 0.15 (0.15) 15 0.05 0.07 0.05 0.20 (0.20) 20 0.06 0.09 0.06 0.25 (0.25) 24 0.02 0.06 0.02 0.30 (0.30) 26 0.00 0.06 0.00 0.35 (0.35) 23 -0.01 0.07 -0.01 0.40 (0.40) 22 -0.01 0.06 -0.02 0.45 (0.45) 26 -0.01 0.07 -0.03 0.50 (0.50) 24 -0.03 0.08 -0.06 0.55 (0.55) 23 -0.04 0.08 -0.08 0.60 (0.60) 19 -0.02 0.08 -0.07 0.65 (0.65) 21 -0.01 0.05 -0.02 0.70 (0.70) 17 0.00 0.08 -0.03 0.75 (0.74) 16 0.04 0.08 0.07 0.80 (0.80) 17 0.05 0.09 0.11 0.85 (0.89) 14 0.08 0.10 0.20 0.90(0.95) 31 0.08 0.10 0.23 0.95 (0.97) 6 0.04 0.06 0.22 1.00 (1.12) 5 0.18 0.30 0.45

Table 4.13 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DOB at GrandHill, MS.

h/H n Residual RMSE Bias 0.00 (0.02) 20 0.02 0.10 0.01 0.05 (0.05) 32 -0.03 0.07 -0.03 0.10 (0.10) 28 0.02 0.03 0.02 0.15 (0.15) 13 0.01 0.02 0.01 0.20(0.20) 20 0.00 0.03 0.00 0.25 (0.25) 23 -0.01 0.03 -0.01 0.30 (0.30) 27 -0.01 0.03 -0.01 0.35 (0.35) 26 0.00 0.02 0.00 0.40 (0.40) 22 -0.01 0.03 0.01 0.45 (0.45) 26 0.01 0.03 0.01 0.50 (0.50) 26 0.00 0.05 -0.01 0.55 (0.55) 29 -0.01 0.04 -0.02 0.60 (0.60) 20 -0.02 0.06 -0.05 0.65 (0.65) 21 -0.01 0.05 -0.04 0.70 (0.70) 17 0.00 0.06 -0.01 0.75 (0.74) 16 0.01 0.07 0.00 0.80 (0.80) 16 0.04 0.07 0.07 0.85 (0.89) 14 0.04 0.08 0.10 0.90 (0.95) 29 0.06 0.10 0.19 0.95 (0.97) 6 0.03 0.05 0.20 1.00 (1.12) 5 0.03 0.30 -0.38

Table 4.14 Residuals, RMSE, and bias of Max and Burkhart taper model for DIB at Poplarville, MS.

h/H n Residual RMSE Bias 0.00(0.01) 12 -0.01 0.09 -0.01 0.05(0.05) 18 0.00 0.04 0.00 0.10(0.10) 16 0.02 0.04 0.02 0.15(0.15) 16 0.00 0.03 0.00 0.20(0.20) 14 0.01 0.03 0.01 0.25(0.25) 18 -0.01 0.05 -0.02 0.30(0.30) 18 -0.02 0.06 -0.03 0.35(0.35) 17 0.00 0.04 -0.01 0.40(0.40) 20 -0.01 0.04 -0.02 0.45(0.45) 19 0.00 0.04 0.00 0.50(0.50) 14 0.02 0.05 0.02 0.55(0.55) 18 0.00 0.06 -0.01 0.60(0.60) 16 0.01 0.08 -0.01 0.65(0.65) 16 0.00 0.07 -0.05 0.70(0.70) 10 0.02 0.08 0.03 0.75(0.75) 11 -0.01 0.08 -0.11 0.80(0.79) 8 0.02 0.09 -0.11 0.85(0.87) 9 -0.04 0.09 -0.79 0.90(0.93) 8 0.03 0.09 0.01 0.95(0.97) 12 -0.01 0.02 -0.66 1.00(0.99) 2 0.00 0.00 -0.14

Table 4.15 Residuals, RMSE, and bias of Cao (2009) model for tallow DIB at Poplarville, MS.

h/H n Residual RMSE Bias 0.00(0.01) 12 -0.03 0.11 -0.03 0.05(0.06) 24 -0.01 0.04 -0.01 0.10(0.11) 11 0.00 0.02 0.00 0.15(0.15) 16 -0.01 0.03 -0.01 0.20(0.20) 13 -0.01 0.03 -0.01 0.25(0.25) 18 -0.03 0.06 -0.04 0.30(0.30) 18 -0.03 0.07 -0.05 0.35(0.35) 17 -0.01 0.05 -0.03 0.40(0.40) 20 -0.02 0.05 -0.04 0.45(0.45) 19 -0.01 0.04 -0.02 0.50(0.50) 14 0.00 0.05 0.00 0.55(0.55) 18 -0.01 0.06 -0.03 0.60(0.60) 16 0.00 0.07 -0.03 0.65(0.65) 16 -0.01 0.07 -0.07 0.70(0.70) 10 0.02 0.08 0.00 0.75(0.75) 11 -0.01 0.08 -0.13 0.80(0.79) 8 0.01 0.09 -0.12 0.85(0.87) 9 -0.04 0.09 -0.81 0.90(0.93) 8 0.03 0.09 0.02 0.95(0.97) 12 -0.01 0.02 -0.64 1.00(0.99) 2 0.00 0.00 -0.04

Table 4.16 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DIB at Poplarville, MS.

h/H n Residual RMSE Bias 0.00(0.01) 12 -0.01 0.11 -0.01 0.05(0.06) 24 0.00 0.04 0.00 0.10(0.11) 11 0.01 0.04 0.01 0.15(0.15) 16 0.00 0.03 0.00 0.20(0.20) 13 0.01 0.04 0.01 0.25(0.25) 19 -0.01 0.05 -0.02 0.30(0.30) 17 -0.01 0.04 -0.02 0.35(0.36) 20 0.00 0.03 0.00 0.40(0.41) 19 -0.01 0.03 -0.01 0.45(0.45) 17 0.00 0.03 0.00 0.50(0.50) 14 0.01 0.04 0.02 0.55(0.55) 18 0.00 0.05 -0.01 0.60(0.61) 17 0.01 0.06 -0.02 0.65(0.65) 16 -0.01 0.06 -0.05 0.70(0.70) 9 0.01 0.09 -0.02 0.75(0.75) 11 -0.01 0.08 -0.10 0.80(0.79) 8 0.00 0.09 -0.16 0.85(0.87) 9 -0.04 0.09 -0.77 0.90(0.93) 8 0.03 0.09 -0.03 0.95(0.97) 12 -0.01 0.02 -0.66 1.00(0.99) 2 0.00 0.01 -0.24

Table 4.17 Residuals, RMSE, and bias of Max and Burkhart taper model for DIB at GrandHill, MS.

h/H n Residual RMSE Bias 0.00 (0.02) 21 0.06 0.10 0.04 0.05 (0.05) 27 -0.06 0.10 -0.07 0.10 (0.10) 22 0.00 0.07 -0.01 0.15 (0.15) 24 0.02 0.05 0.03 0.20 (0.20) 23 0.04 0.05 0.04 0.25 (0.25) 24 0.01 0.05 0.01 0.30 (0.30) 26 0.00 0.04 -0.01 0.35 (0.35) 23 -0.02 0.06 -0.03 0.40 (0.40) 22 -0.01 0.05 -0.03 0.45 (0.45) 26 -0.01 0.06 -0.03 0.50 (0.50) 24 -0.03 0.08 -0.06 0.55 (0.55) 23 -0.04 0.08 -0.09 0.60 (0.60) 19 -0.02 0.09 -0.08 0.65 (0.65) 21 -0.01 0.06 -0.03 0.70 (0.70) 17 0.01 0.08 -0.02 0.75 (0.74) 16 0.04 0.09 0.06 0.80 (0.80) 17 0.05 0.08 0.11 0.85 (0.89) 14 0.08 0.10 0.20 0.90 (0.95) 31 0.07 0.10 0.24 0.95 (0.97) 6 0.03 0.05 0.21 1.00 (1.12) 5 0.20 0.30 0.44

Table 4.18 Residuals, RMSE, and bias of Cao (2009) profile model for tallow DIB at GrandHill, MS.

h/H n Residual RMSE Bias 0.00 (0.02) 21 0.04 0.10 0.03 0.05 (0.05) 33 -0.05 0.08 -0.05 0.10 (0.10) 28 0.00 0.03 0.00 0.15 (0.15) 15 0.04 0.07 0.05 0.20 (0.20) 20 0.06 0.08 0.07 0.25 (0.25) 24 0.02 0.06 0.02 0.30 (0.30) 26 0.00 0.06 0.00 0.35 (0.35) 23 -0.01 0.07 -0.02 0.40 (0.40) 22 -0.01 0.06 -0.02 0.45 (0.45) 26 -0.01 0.08 -0.03 0.50 (0.50) 24 -0.03 0.08 -0.06 0.55 (0.55) 23 -0.04 0.08 -0.09 0.60 (0.60) 19 -0.02 0.08 -0.07 0.65 (0.65) 21 0.00 0.05 -0.02 0.70 (0.70) 17 0.00 0.08 -0.03 0.75 (0.74) 16 0.04 0.09 0.07 0.80 (0.80) 17 0.05 0.08 0.11 0.85 (0.89) 14 0.08 0.10 0.20 0.90 (0.95) 31 0.07 0.10 0.23 0.95 (0.97) 6 0.03 0.05 0.22 1.00 (1.12) 5 0.17 0.30 0.44

Table 4.19 Residuals, RMSE, and bias of Clark, Souter, and Schlaegel taper model for DIB at GrandHill, MS.

h/H n Residual RMSE Bias 0.00 (0.02) 20 0.02 0.10 0.01 0.05 (0.05) 32 -0.03 0.09 -0.03 0.10 (0.10) 28 0.02 0.04 0.02 0.15 (0.15) 13 0.01 0.03 0.01 0.20 (0.20) 20 0.00 0.03 0.00 0.25 (0.25) 23 -0.01 0.03 -0.01 0.30 (0.30) 27 0.00 0.03 -0.01 0.35 (0.35) 26 0.00 0.03 -0.01 0.40 (0.40) 22 0.01 0.04 0.01 0.45 (0.45) 26 0.01 0.04 0.01 0.50 (0.50) 26 0.00 0.06 -0.01 0.55 (0.55) 29 -0.01 0.05 -0.03 0.60 (0.60) 20 -0.02 0.07 -0.05 0.65 (0.65) 21 -0.01 0.05 -0.04 0.70 (0.70) 17 0.00 0.07 -0.02 0.75 (0.74) 16 0.01 0.08 -0.01 0.80 (0.80) 16 0.03 0.08 0.06 0.85 (0.89) 14 0.05 0.09 0.10 0.90 (0.95) 29 0.07 0.10 0.18 0.95 (0.97) 6 0.04 0.06 0.20 1.00 (1.12) 5 -0.03 0.40 -0.36

Figure 4.6 Comparison of the residuals of three models for DOB profile function at Poplarville site (A) and Grand Bay/Sandhill (B), MS, test sites.

Figure 4.7 Comparison of the bias of the three models for DIB profile function at Poplarville site (a) and at Grand Bay/Sandhill (b), MS, test sites.

To compare the fitted DOB and DIB stem profile of tallow at Poplaville versus

Grand Bay and Sandhill, they were plotted using the fitted model with the highest FI:

Clark et al. (1991) model for both tallow at Poplaville and Grand Bay and Sandhill (Fig.

4.7). Still, there was no distinct difference at the bottom part on butt end of the DOB 47

profile, but at the upper section of the stem, tallow at Grand Bay and Sandhill had larger

DOB/DBH at the same height ratio with tallow at Poplaville. However, DIB of tallow at the three sites had the similar trend from bottom to the upper stem; in other words, under the same height ratio, tallow at Poplarville had a smaller diameter ratio compared to tallow at Grand Bay and Sandhill.

Figure 4.8 Best fitted stem profile model for both DOB (A) and DIB (B) of tallow tree at Polplarville (dash) and Grand Bay/Sandhill (solid), MS, respectively

CHAPTER V

DISCUSSION AND COLUSION

5.1 Discussion

Schumacher, Chapman-Richards, logistic, and Mitscherlich models were selected to fit diameter and height growth of tallow in southern Mississippi in different forest ecosystems that included oak/gum/cypress (Poplarville) and longleaf/slash pine forests

(Grand Bay NERR and Sandhill NWR). On the basis of the R square obtained and the capability of describing the ecologic growth process of tallow tree, the logarithmic form of the Schumacher model was the best model for describing DBH/age, height/age, height/DBH, volume/age, and biomass/age. A total of 33 sample tallow trees were used to fit these models which was regarded as reasonable and reliable in statistical inference because the variables estimated in these models was limited. According to the best fitted models, DBH, volume, and biomass of tallow in the oak/gum/cypress forest was greater than that in the longleaf/slash pine forests at the same age; likewise, stem volume and biomass which grew linearly with DBH squares times height, based on Spurr’s volume equation (Bi and Hamilton, 1998) showed that the volume/biomass of tallow grew faster with DBH and height at Grand Bay than that at Poplaville and Sandhill sites. Also comparing the height/DBH model revealed that the height of tallow in the longleaf/slash pine forest was taller than that in oak/gum/cypress forest with the same DBH. The reason for the different growth rate of DBH, height, and volume/biomass for tallow at these 50

three different sites might be caused by the various forest densities; oak-gum-cypress forest had a lower density than that longleaf/slash pine forests. Therefore, the DBH of tallow was larger in the oak-gum-cypress forest while it was taller in the longleaf/slash pine forest resulting in the volume and biomass difference between them.

Previous studies related to exotic tallow tree mainly concerned the spread of tallow in southern forestlands (e.g. Gan et al., 2009; Wang et al., 2011; Fan et al., 2012;

Tan, 2012) and associated facilitating factors (Rogers and Siemann, 2002, 2003; Zou et

al., 2006, 2007, 2008; Pattison and Mack, 2008). Few researchers have paid close

attention to the establishment of tallow in these forestlands. These growth models

developed in this study provide a theoretical basis for managers to estimate and predict

the stocking of tallow in the native forest ecosystems so that corresponding management

decisions may be made. However, due to the limitation of data collection, the models

established here only account for tallow in oak/gum/cypress and longleaf/slash pine

forests in costal Mississippi; hence, to explore the growth rate of tallow in other typical

forest type groups such as oak/pine (Quercus/ Pinus), or oak/hickory (Quercus/ Carya),

more sampling will be necessary. Through comparing growth models for tallow in

different forest type groups, information about stand conditions that best facilitate tallow

growth can be gained. For example, in this study growth rate of DBH, volume, and

biomass of tallow in the oak-gum-cypress forest was more rapid, and it can be concluded

that conditions were more favorable.

For stem profile equations of tallow, three segmented polynomial taper models

including the Max and Burkhart (1976), the Cao (2009) modified Max and Burkhart

model, and Clark et al. (1991) were tested and chosen in this study. Profile datasets were

fitted using TProfile (Matney, 1996) software. Clark et al. (1991) had relatively better performance (RMSE abd FI) for both DOB and DIB profiles of tallow in oak/gum/cypress forest at Poplarville; but there were no significant difference among models. Clark et al. (1991) had a larger FI and smaller RMSE for tallow in longleaf/slash pine at Grand Bay and Sandhill as compared with the Max and Burkhart (1976) and the

Cao (2009) modified Max and Burkhart models. The reason for Clark et al. (1991) model performed better for tallow at the three sites is that this model describes the whole stem with four sectors. Studies (Larson, 1965; Hilt and Dale, 1980; Garber and Maguire, 2003;

Bluhm et al., 2007) have reported that stem profile varied with stand conditions.

However, in this study there is no obvious difference for both DOB and DIB profile of tallow in the two different coastal forests. Leites and Robinson (2004) proved that crown dimensions also affected stem taper, and they improved the taper equations for loblolly pine by incorporating crown length and crown length ratio. The importance of profile equations in improving the estimation of volume/biomass for managing and valuing forests and the difficulty of selecting an appropriate model that works well for multiple species and diverse site conditions makes the equations an exploited research topic

(Clutter et al., 1983; McClure et al., 1986; Muhairwe, 1999).

The Max and Burkhart (1976) model had superior performance in predicting general tree profile because it divided stem into three different sections which joined at two points (Cao, 2009). Nevertheless, Cao’s (2009) model which reduces 10-25% root mean squared error for the upper-stem diameter prediction improves the model precision in estimating upper stem. In this study for both DOB and DIB profiles of tallow at the three study sites, the two models showed similar performance. However, the Max and

Burkhart model was more biased at the upper section of stem. Clark et al. (1991) is a segmented form class model which requires measurement of variable height, total height,

DBH, as well as diameter at Girard form class height. Hence, this model was more accurate (based on RMSE and FI) than the Max and Burkhart (1976) profile model in fitting tallow stem in both oak/gum/cypress and longleaf/slash pine forest. Taper models play an important role in estimating diameter at various heights, tree height with known diameter, and individual log volume in forestry. Nevertheless, due to many factors influencing a stem’s shape, the accuracy and flexibility of taper models still needs further research.

5.2 Conclusions

Through fitting the cumulative growth model DBH/age, height/DBH, volume/age, and biomass/age models, as well as the allometric relationship between stem volume/biomass and DBH and height, invasive tallow tree was shown to be growing exponentially with age in both southern oak/gum/cypress and longleaf/slash pine forests.

Also, the height of individual tree responded differently to stand conditions associated with the two distinct forest types. At the same age, DBH, volume, and biomass of tallow in the oak/gum/cypress forest was greater than tallow in longleaf/slash pine forest. For instance, the DBH, volume, and biomass of 10-year old tallow in oak-gum-cypress forest was 18.72 cm, 0.134 m3, 29.3 kg while in the longleaf/slash pine forest was 14.5 cm, 0.07 m3, and 16.5 kg, respectively. However, the height of tallow grew faster with DBH in the

longleaf/slash pine forest than in the oak-gum-cypress forest. Based on the height/DBH

model, with a DBH of 10 cm, the height of tallow was computed to be 10.7 m in the

longleaf/slash pine forest while it was 9.4 m in the oak-gum-cypress forest. By contrast, 53

there was no significant difference between the height/age and volume/DBH models for tallow in the oak-gum-cypress forest and longleaf/slash pine forests. Height grew exponentially with age and volume changed exponentially with DBH. Additionally, the linear volume/biomass model with DBH squared times height showed that volume/biomass grew faster at the Grand Bay site than it did at the Poplaville and

Sandhill sites under the same value of DBH square times height.

For stem profile, there was no significant difference between the DOB profile of tallow in the oak/gum/cypress forest and the longleaf/slash pine forest based on the results of fitting three segmented profile models. Nevertheless, the DIB profile of tallow at Poplarville (Fig.4.7 B) showed that the DIB/DBH ratio was smaller than that at the

Grand Bay and Sandhill sites under the same height ratio. This difference might be caused by the different bark thickness of tallow in the oak-gum-cypress forest as

compared to longleaf/slash pine forest, but further research is needed for verification.

Tallow tree in the longleaf/slash pine forest were slender as compared to the less slender

tallow in the oak-gum-cypress forest at Poplaville as observed and shown by the

height/DBH model and profile equations. The height/DBH model showed that for the

same DBH, tallow was taller in the longleaf/slash Pine forest while taper function

demonstrated that with the same DOB (DIB)/DBH ratio, tallow in longleaf/slash pine

forest had a greater height ratio.

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APPENDIX A

FIELD AND LAB IMAGES

Figure A.1 Fell down the sample tallow trees in sample sites

Figure A.2 Separate the tree stem into 1 meter sections and abstract disks from the middle of each section

Figure A.3 Measure the volume of each disk with water displace method

Figure A.4 The sander used to sand all disks before counting rings

Figure A.5 Method of sanding all disks with the equipment before getting age of each tree

Figure A.6 Counting rings on each disk with pins marking them