# Introduction

Sissoo is known as a premier timber species of the rosewood genus with the common name sissoo in Bangladesh. It is native to Sub-Himalayan zone including India, Pakistan and Afghanistan ([35], [20]). It is also recognized as an important species for fuel wood, shade tree, agroforestry and fodder in the region ([35], [34], [13]). Foresters traditionally derive timber volumes by employing allometric techniques. Allometric relationship for estimating stand volume as well as forest biomass is very important for managing any natural and artificial forest resources ([5], [10], [22], [24]). In the estimation of stand volume, the uncertainties appear due to the measurements of basal area and the use of a mean form factor, which is related to mean tapering ([24]). Thus, allometric relationships offer better estimates of the forest standing volume, which is also an important parameter in further research such as biomass and carbon-emission estimates avoiding the uncertainties in the bole-volume estimates ([16], [24]). Therefore, choosing a suitable functional variable in the allometric equation is very important for allometric techniques in forest science ([16], [17]).

There are various independent variables in the allometric relationships to estimate biomass. In most studies, *D* (dbh, stem diameter at 1.3 m height above the ground) was taken as the only independent variable in the allometric equation (*e.g.*, [23], [29], [12], [1], [11], [26]). However, incorporation of the variable *H* (tree height - *i.e.*, the use of *D*^{2}*H*) ensures higher accuracy of allometric estimation in some tree species ([32], [33], [18], [28]). Moreover, the use of the new variable *D*_{0.1}^{2}*H* (*D*_{0.1}, diameter at one-tenth of *H*) instead of *D*^{2}*H* has been suggested to improve the accuracy of estimation ([25], [14], [17]). Attiwill ([3]) found a strong allometric relationship between the girth (at the point immediately before branching) of main branches of a tree and branch dry weight in a stand of *Eucalyptus obliqua* L’ Herit. The use of *D*_{B} (stem diameter at a height of clear bole length) provides better results in estimating the weight of branch and leaf, and leaf area per tree, as described by the pipe model theory of Shinozaki et al. ([30]). Various allometric equations have been developed from different tropical species ([7], [27], [6], [2], [9], [16], [10]). It is evident that mainly species of dicotyledonous trees differ in allometry because of distinctive bole shape. The allometric equations developed from various species yield useful estimates for large-scale inventories.

In this paper, we seek to establish the allometric relationships of the stem volume of individual trees to different dimensions, such as *D*, *D*^{2}, *D*^{2}*H*, *D*_{0.1}, *D*_{0.1}^{2} and *D*_{0.1}^{2}*H* and to propose a standard method for predicting the stem volume of Sissoo.

# Materials and methods

## Study site and species description

The study was carried out in block plantations at Khulna located in the southern part of Bangladesh (Fig. 1). The area belongs to subtropical region. *Dalbergia sissoo* is one of the important plantation species in the region.

The species is a strong-light demander and shows good coppicing ability ([35]). The development of seedling is better in full overhead light than under partial shade. It is capable of growing under adverse soil and moisture conditions ([34]). Though, *D. sissoo* is a frost hardy species, its young leaves are adversely affected by severe frost and even the poles get completely defoliated ([31]). It has a complex root system that consists of a deep tap root and long lateral roots. Root suckers are produced from the lateral roots. Root suckers develop towards the end of the rainy season and may attain a height of more than 2 m on good soil ([15]). The species can grow naturally up to 1500 m elevation. The temperature in its native range averages 12-22 °C, but varies from just below freezing to nearly 50 °C with an average annual rainfall of 500 to 2000 mm with a monsoonal pattern and drought period of 3-4 months ([35]). The growth of the bole is crooked and straight logs of any great length are difficult to obtain. The heartwood is brown with darker streaks and because of its strength and durability, the wood is highly valued for furniture, constructional and general utility purposes.

## Sampling, data collection and analysis

All the studies were carried out in May 2006 in a monoculture plantation of *D. sissoo* where the canopy was completely closed. A destructive sampling of 30 individual trees with a wide range of diameter and height were used for this study (Tab. 1). The following measurements were carried out: tree height *H*, stem diameter at a height of *H*/10 (*D*_{0.1} - [17]), stem dbh *D* and stem diameter at 1.0 m interval thereafter up to the merchantable height (stem height at 10 cm diameter). For measuring the diameter, girth values were converted to diameter after divided with π. Stem volume was calculated using the Smalian’s formula ([4]).

**Tab. 1 -**Description of sissoo sample trees used for this study.

*H*: Tree height;

*D*

_{0.1}: stem diameter at a height of

*H*/10;

*D*: stem diameter at 1.3 m height (dbh);

*V*: stem volume.

Tree No. | D (cm) | D_{0.1} (cm) | H (m) | V (cm^{3}) |
---|---|---|---|---|

1 | 9.549 | 10.027 | 9.01 | 10403.4 |

2 | 9.708 | 10.504 | 8.01 | 17653.9 |

3 | 10.134 | 11.141 | 8.02 | 25678.3 |

4 | 10.663 | 10.982 | 9.01 | 20521.2 |

5 | 10.759 | 11.459 | 8.07 | 27829.2 |

6 | 11.141 | 11.937 | 9.03 | 14610.4 |

7 | 11.513 | 12.414 | 7.50 | 36859.9 |

8 | 11.678 | 13.051 | 8.02 | 37520.3 |

9 | 11.937 | 11.937 | 9.75 | 38089.3 |

10 | 12.321 | 12.573 | 8.50 | 41628.1 |

11 | 12.614 | 12.892 | 8.50 | 46894.1 |

12 | 12.984 | 12.796 | 12.01 | 51629.2 |

13 | 13.210 | 13.528 | 11.03 | 62953.1 |

14 | 13.242 | 13.210 | 11.02 | 56349.9 |

15 | 13.687 | 13.433 | 10.25 | 57361.4 |

16 | 13.866 | 14.006 | 10.25 | 36131.8 |

17 | 14.006 | 14.324 | 8.50 | 60054.8 |

18 | 14.961 | 14.801 | 14.01 | 89369.7 |

19 | 15.597 | 15.756 | 14.04 | 101757.4 |

20 | 16.470 | 16.870 | 14.02 | 128149.9 |

21 | 16.999 | 17.666 | 11.01 | 139271.9 |

22 | 17.189 | 17.507 | 10.75 | 122993.6 |

23 | 21.963 | 22.282 | 12.50 | 206910.6 |

24 | 26.897 | 26.420 | 15.10 | 411254.8 |

25 | 27.056 | 26.420 | 15.00 | 390567.9 |

26 | 27.693 | 27.693 | 14.75 | 437078.1 |

27 | 29.155 | 28.254 | 20.05 | 569421.1 |

28 | 31.210 | 29.155 | 20.02 | 665874.1 |

29 | 32.675 | 30.152 | 20.50 | 759421.1 |

30 | 33.423 | 31.831 | 21.75 | 819421.1 |

The simple allometric equation is generally written using the power curve ([17]) in the form (eqn. 1):

where *y* is the dependent variable and *x* is the independent variable, and *a* the coefficient and *b* the allometric constant. The equation is linearized by taking logarithms, as follows (eqn. 2):

where ln *a* and *b* are the intercept and slope of the regression line, respectively. The ln *a* and *b* are obtained by the method of least squares. In this study, the allometric relationships of the volume and different dimensions such as *D*, *D*^{2}, *D*^{2}*H*, *D*_{0.1}, *D*_{0.1}^{2} and *D*_{0.1}^{2}*H* were also established using following equations (eqn. 3, eqn. 4, eqn. 5, eqn. 6, eqn. 7):

The coefficient of determination *R*^{2} was calculated using the following equation (based on the real data before logarithmic transformation - eqn. 8):

where *y*_{i} is the observed value, *y*_{ei} is the corresponding values calculated from the regression line, and *y*_{mi} is the mean of the observed values ([19]). The *R*^{2} value (coefficient of determination) is a measure of the goodness-of-fit between the observed and calculated values ([17]).

# Results

Various allometric equations were developed for data fitting. The allometric relationships of stem volume of sissoo trees to *D* and* D*^{2} are illustrated in Fig. 2. The scatter plot shows a non-linear trend when *D* is used as independent variable. This trend is changed to linear distribution if *D*^{2} is used (Fig. 2). This is also illustrated by the coefficient of determination using *D*, where the power equation (*R*^{2} = 0.970) shows better fitting than linear equation (*R*^{2} = 0.944). When *D* values are squared, the linear equation shows stronger relationship (*R*^{2} = 0.983) than power equation (*R*^{2} = 0.970 - Tab. 2). In this case, the polynomial cubic equation showed the best fit for both *D* (*R*^{2} = 0.997) and *D*^{2} (*R*^{2} = 0.996) with a very close estimate by the quadratic equation for *D* (*R*^{2} = 0.993) and *D*^{2} (*R*^{2} = 0.996).

**Tab. 2 -**Summarized coefficients of the relationships between individual tree volumes of Sissoo to different independent variables.

*H*: Tree height;

*D*

_{0.1}: stem diameter at a height of

*H*/10;

*D*: stem diameter at 1.3 m height (dbh). The units:

*D*= [cm], D

_{0.1}= [cm],

*H*]= [m]. LIN = Linear, LOG = Logarithmic, QUA = Quadratic, CUB = Cubic, POW = Power, EXP = Exponential.

Variable | Equation | a | b | c | d | R^{2} | F | Sign. |
---|---|---|---|---|---|---|---|---|

D | LIN | -342298 | 30629.2 | - | - | 0.944 | 474.4 | < 0.01 |

LOG | -1E+06 | 561873 | - | - | 0.869 | 186.3 | < 0.01 | |

QUA | 145717 | -24827 | 1322.63 | - | 0.993 | 1925.9 | < 0.01 | |

CUB | -203182 | 34316.7 | -1754.7 | 49.327 | 0.997 | 2588.8 | < 0.01 | |

POW | 15.91 | 3.108 | - | - | 0.97 | 902.6 | < 0.01 | |

EXP | 5528 | 0.159 | - | - | 0.928 | 361.7 | < 0.01 | |

D^{2} | LIN | -75972 | 739.7 | - | - | 0.983 | 1660.3 | < 0.01 |

LOG | -1E+06 | 280937 | - | - | 0.869 | 186.3 | < 0.01 | |

QUA | -13315 | 317.6 | 0.3825 | - | 0.996 | 3517 | < 0.01 | |

CUB | -31764 | 487.7 | 0.0288 | 0.0002 | 0.996 | 2462.9 | < 0.01 | |

POW | 15.91 | 1.554 | - | - | 0.97 | 902.6 | < 0.01 | |

EXP | 23664 | 0.0036 | - | - | 0.867 | 182.3 | < 0.01 | |

D_{0.1} | LIN | -390344 | 33385.1 | - | - | 0.925 | 343 | < 0.01 |

LOG | -2E+06 | 611406 | - | - | 0.86 | 172.4 | < 0.01 | |

QUA | 275457 | -42005 | 1844.01 | - | 0.981 | 696.3 | < 0.01 | |

CUB | -445256 | 80012.7 | -4601.5 | 106.354 | 0.989 | 782.3 | < 0.01 | |

POW | 7.04 | 3.385 | - | - | 0.961 | 695.9 | < 0.01 | |

EXP | 4172 | 0.1752 | - | - | 0.928 | 362.9 | < 0.01 | |

D_{0.1}^{2} | LIN | -99651 | 828.7 | - | - | 0.964 | 742.6 | < 0.01 |

LOG | -2E+06 | 305703 | - | - | 0.86 | 172.4 | < 0.01 | |

QUA | 7859 | 104.7 | 0.7173 | - | 0.987 | 1043.7 | < 0.01 | |

CUB | -30223 | 453.3 | -0.0564 | 0.0005 | 0.988 | 721.3 | < 0.01 | |

POW | 7.04 | 1.693 | - | - | 0.961 | 695.9 | < 0.01 | |

EXP | 20529 | 0.0042 | - | - | 0.881 | 207.6 | < 0.01 | |

D_{2}H | LIN | -5209 | 34.9 | - | - | 0.995 | 5738.1 | < 0.01 |

LOG | -1E+06 | 205386 | - | - | 0.877 | 199 | < 0.01 | |

QUA | -15721 | 40.2 | -0.0002 | - | 0.997 | 4000.6 | < 0.01 | |

CUB | -17734 | 41.7 | -0.0004 | 4.90E-09 | 0.997 | 2584.4 | < 0.01 | |

POW | 11 | 1.125 | - | - | 0.959 | 656.2 | < 0.01 | |

EXP | 35224 | 0.0002 | - | - | 0.785 | 102.3 | < 0.01 | |

D_{0.1}^{2} H | LIN | -18747 | 39.3 | - | - | 0.995 | 5791.8 | < 0.01 |

LOG | -2E+06 | 219854 | - | - | 0.877 | 199.2 | < 0.01 | |

QUA | -20488 | 40.2 | -5.00E-05 | - | 0.995 | 2811.9 | < 0.01 | |

CUB | -10661 | 33 | 0.0008 | -3.00E-08 | 0.996 | 1959.7 | < 0.01 | |

POW | 5.65 | 1.204 | - | - | 0.959 | 654.2 | < 0.01 | |

EXP | 32624 | 0.0002 | - | - | 0.808 | 118.1 | < 0.01 |

Fig. 3 illustrates the allometric relationships of stem volume to *D*_{0.1} and *D*_{0.1}^{2}. As observed with the variable *D*, the use of *D*_{0.1 } alsoshowed strong data fitting (*R*^{2} = 0.925) in the allometry (Tab. 2). This relationship is further improved (*R*^{2} = 0.964) when the *D*_{0.1} value is squared. The power equation for both the variables *D*_{0.1} and *D*_{0.1}^{2} showed the same coefficient of determination (*R*^{2} = 0.961). For both the variables *D*_{0.1} and *D*_{0.1}^{2} the polynomial cubic and quadratic equations showed a slight stronger fitting (Tab. 2) in comparison with other equations.

As illustrated in Fig. 4, the incorporation of tree height *H* in the allometric equation gives a better fitting in the linear equation, specially for both *D*^{2}*H* (*R*^{2} = 0.995) and *D*_{0.1}^{2}*H* (*R*^{2} = 0.995). The polynomial cubic along with the quadratic equation showed a very close fit in comparison with the linear equation for both variables *D*^{2}*H* and *D*_{0.1}^{2}*H* (Tab. 2).

# Discussion

Although the allometric equations were highly significant (*P* < 0.01) there was considerable variation among them as indicated by the coefficient of determination *R*^{2} values (Tab. 2). The scatter plotting (Fig. 2) shows a non-linear trend for *D* as independent variable, which becomes linear when plotted against *D*^{2}. This indicates that tree volume is more correlated with basal area than with simple dbh ([8]). Using the simple *D* as independent variable in the allometric equation, the cubic equation showed the best fit (*R*^{2} = 0.997) with a very close estimate by the quadratic equation (*R*^{2} = 0.993). However, there were low differences in the goodness-of-fit among the polynomial, power and linear equations. As the quadratic and cubic equations consist of several coefficients, for practical applications in stand volume estimation, because of simplicity, the linear or power equations the use of *D*^{2} as an independent variable should be preferred ([17]).

Like the commonly known variable *D*, the use of *D*_{0.1 } also showed strong linear data fitting (*R*^{2} = 0.925) in the allometry (Tab. 2). This degree of linearity was further improved (*R*^{2} = 0.964) when *D*_{0.1}^{2} value is used instead of *D*_{0.1} ([14], [17]). Here, the cubic equation showed the best fit for both *D*_{0.1} (*R*^{2} = 0.989) and *D*_{0.1}^{2} (*R*^{2} = 0.988). The next strong fit is also from the quadratic equation for *D*_{0.1} (*R*^{2} = 0.981) and *D*_{0.1}^{2} (*R*^{2} = 0.987). Overall, it may be remarked that the allometric relationships of stem volume to the tree diameter at 10% of tree height (*D*_{0.1}) did not improve the allometric strength in *D. sissoo* in comparison with simple *D*, as reported in case of some tree species ([14], [17]).

The multiplication of tree height *H* with diameter or basal area in the allometric equation gives high degree of linearity for both the variables* D*^{2} *H* (*R*^{2} = 0.995) and *D*_{0.1}^{2}*H* (*R*^{2} = 0.995) in the allometric estimation. This suggests that biologically tree diameter and height change proportionality with the change of tree size ([17]). Hence, *H* is incorporated in the allometric equations, the polynomial cubic and quadratic equations showed a similar degree of fitting in comparison with linear equation for both the variables *D*^{2}*H* and *D*_{0.1}^{2}*H*, because of simplicity, the linear equation would be preferred for indirect estimation in the field with a good level of accuracy (*R*^{2} = 0.995).

For predicting timber yield ([21]) foresters often combine trunk diameter and height measurements ([21], [4]) as the independent variables in allometric relationships. However, for the* D. sissoo* plantations studied the stem dbh alone showed a very strong accuracy of estimation (*R*^{2} = 0.983 to 0.997) especially when used as *D*^{2}. Thus, it is concluded that the use of tree height in the allometric equation ([32], [18], [28], [17]) can be neglected for *D. sissoo*, as far as the present study area is concerned. Therefore, for estimating the stem volume of sissoo, the use of *D*^{2} as an independent variable in the allometric equation with a linear or quadratic equation is recommended.

The findings of this study indicate that there is a variation in the use of independent variables in allometric equations for estimating the stem volume of the species. The allometric relationships described in this paper may not be appropriate in mixed or open forest stands, because the present study was carried out under monospecific and closed canopy conditions. For estimation stem volume of trees outside the size range of this investigation, care should be taken in extrapolating the present allometric relationships. Therefore, users of these allometric equations are recommended to check some individual trees outside the present size class.

# Acknowledgements

We are grateful to Forestry and Wood Technology Discipline, Khulna University, Bangladesh for providing logistic support for the field data collection. The data analysis and manuscript preparation were performed in the Institute of Forest Growth and Forest Computer Sciences, Technische Universität Dresden, Germany, which was supported by the Alexander von Humboldt Stiftung / Foundation, Germany.

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