Annual Tree-Ring Curve-Fitting for Graphing the Growth Curve and Determining the Increment and Cutting Cycle Period of Sungkai (Peronema canescens)

Abstract

Growth and increment are extremely important in sustainable forest management, and in forest inventory they are periodically measured in a permanent sampling unit. The age of a tree is often unknown, especially in natural, community, and urban forests; therefore, determining growth and increment can be problematic. The aim of this study was to propose a solution for this problem by conducting annual tree-ring curve-fitting to determine a tree’s age-related dimension so that growth and increment can then be calculated smoothly. Sungkai (Peronema canescens), a luxurious commercial timber chosen as a case study, resulted in a satisfying growth curve following continuous models (Gompertz, Chapman–Richards, and von Bertalanffy) and discrete models (Bahtiar and Darwis exponential modification). The Chapman–Richards model gave the best-fit sigmoid growth curve. The first derivation (dN/dt) of the growth formula produces the current annual increment (CAI). CAI intersection with mean annual increment (MAI) at the peak of MAI resulted in the optimum biological rotation age and a cutting cycle period of 30 years for the Sungkai plantation commonly planted in urban forests.

1. Introduction

Sungkai (Peronema canescens Jack, synonymous with Peronema heterophyllum Miq.) is a Verbenaceae family member [1] commonly grown as a pioneer tree in secondary forests and it currently belongs to the ITTO’s lesser-used tree species and unrestricted class [2]. Its native distribution area is West Java, Kalimantan, Sumatra, the Riau Archipelago, and Peninsular Malaysia [3]. Sungkai plantation occurs in Indonesia [4] and Malaysia [5]. The Indonesian Ministry of Environment and Forestry recommends this tree species for planting in industrial plantation forests (Hutan Tanaman Industri, (HTI)) for reforestation to increase the productivity of shrubs and in widespread former shifting cultivation lands. Oil palm estates may also adequately plant Sungkai for agroforestry [6] to increase their biodiversity and improve ecosystem expediency. People living in villages also commonly plant Sungkai for shade trees, live fences, and decorative plants in gardens, backyards, and urban forests. Although its growth is slightly slower than other fast-growing species (i.e., Acacia mangium, Albizia falcataria, and Eucalyptus grandis), Sungkai offers several significant benefits [7] such as (1) vegetative propagation by stem cutting [8] can complement its generative seedling regeneration [9]; therefore, it does not depend on the flowering and fruiting season; (2) its tenacity enables planting in a wide range of environmental condition; (3) most parts of the tree (leaves, bark, root, and wood) are traditionally useful for medicine, pharmacology, and as phytoconstituents [10,11,12]; and (4) its timber is high quality and luxurious.

Sungkai is high-quality commercial timber for general housing, furniture and cabinets, plywood and veneer, containers, truck bodies, and truck flooring; its characteristics include specific gravity 0.58, air-dry density 0.64 g/cm³, modulus of rupture (MOR) 660 kgf/cm², modulus of elasticity (E) 83,083 kgf/cm², maximum crushing strength (MCS) 317 kgf/cm², and shear strength 56 kgf/cm² [2]. Combined ring-porous and semi-ring-porous anatomical characteristics with distinct growth rings saliently produce boards of beautiful natural yellow or light brown gradation patterns. This artistic wood has an indistinct sapwood–heartwood transition. The grain is generally straight and occasionally wavy. The texture ranges from moderately fine to moderately coarse. The natural durability class is 3 (moderately durable). Preservatives can easily penetrate the timber, but its air-drying process requires a fairly long time. Woodworkers prefer working with Sungkai for carpentry because they encounter less difficulty cutting it using a hand- or power-saw and during further processing using a planner, sander, chiseling, drilling, or molding machine. It can also be glued easily to make connections between component members. Because of its unique characteristics, Sungkai is traded as a luxurious timber species for multiple usages, such as high-quality veneer, cabinets and carvings [13], decorative plywood [14], exhibiting part of a building (flooring and wall), handicraft, sculpture, ornament, furniture, and interior design.

The demand for Sungkai timber is increasing with the wealthier economy and the developing cultural perception of sustainability and environmentally friendly insight; thus, a mass and intensive plantation is necessary to equilibrate the supply and demand. A sustainable forest management system must be implemented to meet this need in the long-term for present and future generations. Wood production relates to tree growth and increment. The growth course of a tree manifests its genetics, fitness, and competitiveness in its habitats, and it intercorrelates to the population dynamics and stand structure. The sizes (dimensions) of a tree are the function of its age (N = f(t)), and they can be plotted in a diagram as a growth curve. The growth curve relates to time series data and can be composed continuously or discretely based on periodic and regular measurements. The first derivation of the growth curve (dN/dt) indicates the current annual increment (CAI) if the time (t) unit is years, while the ratio of the current dimension (N_i) to its age (t_i) is the mean annual increment (MAI). In this study, the tree growth curve is discussed to enrich the knowledge of managing wood production sustainability by equalizing the current annual increment (CAI) and mean annual increment (MAI). When CAI and MAI curves intersect, the current increment is equal to the average. The CAI of juvenile stands is higher than its MAI, while older ones may have a CAI smaller than the MAI.

Forest inventory systematically and periodically measures tree dimensions to provide reliable data for management decisions related to silvicultural treatment for tree improvement and to estimate timber production. It assesses and evaluates the stands’ current status to provide a basis for analysis and planning, building a foundation for sustainable forest management [15]. Tree age-related dimension databases are usually accumulated from periodic dimension measurements of permanent sample units [16] in certain plots. The data are plotted in a diagram to graph and curve-fit the growth curve based on the empirically and scientifically justified equation models.

The precise age of a tree is often unknown, especially in natural forests where the seedling grew without obvious plantation. In villages, people rarely record their sowing or the plantation years of the urban or community forests. In cases of ancestral planting, descendants may not know the age of inherited trees. In this study, tree annual growth rings are proposed to represent the age-related dimension as an alternative to periodic forest inventory measurement. The distinct environmental conditions (i.e., sunray intensity, water supply, temperature, and relative humidity) during rainy–drought seasons affect the cambium differentiation-producing xylem of earlywood and latewood. The gradation of earlywood and latewood generates the growth rings, and their numbers counted from the pith may represent a tree’s age.

Denih et al. [17] proposed the model in Equation (1) to curve-fit raw data resembling an elliptical shape located in an arbitrary position in a polar coordinate system. Since it has been proven reliable for curve-fitting a tree stem’s cross-sectional shape, it can also potentially estimate the annual tree ring shape and location. Non-linear regression estimates all parameters included in the model (a, b, r_o, θ_o, and k) to determine the best-fit curve. Then, the estimated parameters representing tree dimension become the ordinates (y) plotted versus tree ring count numbers representing the tree ages as abscissa (x). The growth functions (i.e., Gompertz [18], Chapman–Richards [19,20,21], von Bertalanffy [22], and exponential modifications [23,24]) fit the plot of tree age (x) vs. estimated dimension (y) to draw the growth curve.

$$r_i=\frac{r_o\left(\left(a^2-b^2\right)\sin \left({\mathsf{\theta }}_i-k\mathsf{\pi }\right)\sin \left({\mathsf{\theta }}_o-k\mathsf{\pi }\right)+b^2\cos \left({\mathsf{\theta }}_i-{\mathsf{\theta }}_o\right)\right)\pm ab\sqrt{\left(a^2-b^2\right){\sin }^2\left({\mathsf{\theta }}_i-k\mathsf{\pi }\right)+b^2-{r_o}^2{\sin }^2\left({\mathsf{\theta }}_i-{\mathsf{\theta }}_o\right)}}{\left(a^2-b^2\right){\sin }^2\left({\mathsf{\theta }}_i-k\mathsf{\pi }\right)+b^2}+{\mathsf{ℰ}}_i$$

(1)

where: (a, b) is the semi-major and semi-minor, (r_o, θ_o) is the ellipse’s center in polar form, and kπ is the rotation angle about the ellipse’s center.

In this study, the best-fit growth curve was chosen and its first derivation was solved to draw the CAI curve. In order to obtain the objective of determining the cutting cycle period, the intersection between CAI and MAI curves was calculated. The MAI curve is the growth curve divided by the corresponding age. This intersection equilibrates the timber productions and costs, and management should consider it to decide the silvicultural action on the stands. A well-planned cutting cycle period based on biological rotation age analysis may significantly contribute to a sustainable forest management system.

2. Materials and Methods

The study specimen was a disc taken from the National Board of Research and Innovation (Badan Riset dan Inovasi Nasional, (BRIN))’s collection. The disc was cut from a 26–28-year-old Sungkai (Peronema canescens Jack) tree at breast height (approximately 130 cm). The disc was sanded to display each annual ring clearly. The disc’s cross-section was scanned together with a perpendicular ruler. The photograph was scaled into its real measurement (1:1), then every annual growth ring was digitized using Webplot Digitizer version 4.6 application software (https://automeris.io/WebPlotDigitizer/) accessed on 22 June 2023. Sungkai tree rings are clearly visible; thus, the anatomical characteristic related to the growth stripe interval [25] was not assessed in determining the existence of each tree ring.

Use of Webplot Digitizer resulted in a comma-separated values (*.csv) file that contained the coordinate of every point in a rectangular (Cartesian) coordinate system (x,y); therefore, transformation into a polar coordinate system (r,θ) [17] was necessary to apply non-linear regression following the Equation (1) model. For simplicity, the center of the first ring translated as moving coincide with the pith position; therefore, the center of the first annual tree ring is the origin (0,0). The transformation into a polar coordinate system is given in the following Equations (2) and (3). Since Microsoft Excel’s asin formula results in a value between −π/2 ≤ θ ≤ π/2, the following Equation (4) modification is necessary for θ values outside this range.

$$r=\sqrt{x^2+y^2}$$

(2)

$$\mathsf{\theta }=\mathrm{asin}\left(\frac{y}{r}\right)$$

(3)

$$\mathsf{\theta }=\left\{\begin{array}{l}\begin{array}{l}\mathrm{asin}\left(\frac{y}{r}\right) ;\mathrm{for} 0\le \mathsf{\theta }\le \frac{\mathsf{\pi }}{2}\\ \mathsf{\pi }-\mathrm{asin}\left(\frac{y}{r}\right) ;\mathrm{for} \frac{\mathsf{\pi }}{2}\le \mathsf{\theta }\le \mathsf{\pi }\end{array}\\ \begin{array}{l}\mathsf{\pi }-\mathrm{asin}\left(\frac{y}{r}\right) ;\mathrm{for} \mathsf{\pi }\le \mathsf{\theta }\le \frac{3\mathsf{\pi }}{2}\\ 2\mathsf{\pi }+\mathrm{asin}\left(\frac{y}{r}\right) ;\mathrm{for} \frac{3\mathsf{\pi }}{2}\le \mathsf{\theta }\le 2\mathsf{\pi }\end{array}\end{array}\right.$$

(4)

Although Levenberg–Marquardt [26,27,28,29] or Gauss–Newton [30,31,32] iterations may result in true (convergent) values to predict all parameters both for plus (+) and minus (−) signs of Equation (1), we prefer to choose the plus (+) sign model (Equation (5)) for conducting the ellipse curve-fitting using non-linear regression.

$$r_i=\frac{r_o\left(\left(a^2-b^2\right)\sin \left({\mathsf{\theta }}_i-k\mathsf{\pi }\right)\sin \left({\mathsf{\theta }}_o-k\mathsf{\pi }\right)+b^2\cos \left({\mathsf{\theta }}_i-{\mathsf{\theta }}_o\right)\right)+ab\sqrt{\left(a^2-b^2\right){\sin }^2\left({\mathsf{\theta }}_i-k\mathsf{\pi }\right)+b^2-{r_o}^2{\sin }^2\left({\mathsf{\theta }}_i-{\mathsf{\theta }}_o\right)}}{\left(a^2-b^2\right){\sin }^2\left({\mathsf{\theta }}_i-k\mathsf{\pi }\right)+b^2}+{\mathsf{ℰ}}_i$$

(5)

Every annual tree ring’s best-fit curve has semi-major (a) and semi-minor (b) values; thus, its diameter (D) can be estimated with Equation (6), and its area following ellipse and circle approaches can be estimated with Equations (7) and (8).

$$D=a+b$$

(6)

$$A_e=\mathit{\pi }ab$$

(7)

$$A_c=0.25\mathit{\pi }{D}^2$$

(8)

Krisnawati and Bustomi [33] conducted a forest inventory in Banten, Java island. They proposed an allometric formula to predict the clear-bole height (H) and volume (V₁ and V₂) of Sungkai following Equations (8)–(10), where the diameter (D), height (H), and volume units are cm, m, and m³, respectively. Those allometric equations are suitable for Sungkai plantation in urban and community forests.

$$H=2.3094D^{0.4122}$$

(9)

$$V_1=0.00024D^{2.08}$$

(10)

$$V_2=0.0001D^{1.7}{H}^{1.01}$$

(11)

Gompertz (Equation (12)), Chapman–Richards (Equation (13)), and von Bertalanffy (Equation (14)) functions are continuous formulas employed in this study for curve-fitting the growth of diameter, area, and volume.

$$N=a\exp \left(-b\exp \left(-\left(c^t\right)\right)\right)$$

(12)

$$N=a{\left(1-b\exp \left(-Kt\right)\right)}^{\frac{1}{1-m}}$$

(13)

$$N=a\left(1-\exp \left(-K\left(t-b\right)\right)\right)$$

(14)

where:

N = dimension (diameter (D, cm), area (A_e or A_c, cm²), and volume (V₁ or V₂, m³)),

t = tree age (year), or counted ring number from the pith,

a, b, c, and K = estimated parameters.

Bahtiar and Darwis [23] proposed exponential modification to fit growth curves using discrete methods following the Equation (15) basic concept. Equation (15) defines that increment (the difference between the next and current dimensions) is the function of the current dimension multiplied by the age difference. This transformation method hinders the non-linear regression but runs linear, logarithmic, or polynomial regression; therefore, a spreadsheet or standard scientific calculator can solve it by rephrasing Equation (15) as Equation (16a). Since delta t (dt) is the age difference (t_i+1 − t_i), Equation (16b) arises and becomes the model for linear or polynomial regression, where the response (dependent) variable is $\frac{N_{i+1}-N_i}{N_i\left(t_{i+1}-t_i\right)}$ and the predictor (independent) variable is N_i.

$$N_{i+1}-N_i=N_{i}f\left(N_i\right)dt$$

(15)

$$\frac{N_{i+1}-N_i}{N_{i}dt}=f\left(N_i\right)$$

(16a)

$$\frac{N_{i+1}-N_i}{N_i\left(t_{i+1}-t_i\right)}=f\left(N_i\right)+{\mathsf{ℰ}}_i$$

(16b)

The goodness of fit of each regression analysis was measured using its coefficient of determination (R²) and mean square error (MSE). The R² and MSE are familiar in simple linear [34,35,36,37,38,39], multiple linear [40,41,42,43], and non-linear [17,44,45,46,47,48,49] regression to determine its goodness of fit. Higher R² and smaller MSE indicate the better-fit model. The best-fit curve was chosen for further processing to determine the increment. The current annual increment (CAI), sometimes called growth velocity, is the first derivation of the growth formula (Equation (17)), while MAI is the ratio of growth to age (Equation (18)). The intersection of CAI and MAI curves, then solved for tree age (t), is the cutting cycle period (Equation (19)). After this intersection period, the CAI is smaller than MAI. The timber harvesting in the proposed cutting cycle period aims to maintain CAI higher than MAI. Since the disc specimen was cut from the stem at breast height (130 cm), the time needed by the tree to grow to reach this height shall be added to propose the cutting cycle period. Sungkai needs approximately two years to reach 130 cm height; thus, the proposed cutting cycle period is the CAI and MAI intersection plus two years.

$$CAI=\frac{dN}{dt}$$

(17)

$$MAI=\frac{N_i}{t_i}$$

(18)

$$\frac{1}{N}dN=\frac{1}{t}dt$$

(19)

3. Results and Discussion

3.1. Annual Ring

The gradual bands of earlywood and latewood produced by cambium differentiation in the rainy and drought seasons generate tree rings that reflect a tree’s life history. The annual tree-ring topic recently became popular in transdisciplinary research. This novel tool continuously facilitates various analytical techniques to provide methods to answer multidisciplinary scientific problems. Annual tree rings provide signals which can be analyzed for dating and reconstructing the past environmental condition when the tree lived [50,51,52,53,54,55]. The science community collects tree-ring datasets into national repositories, exemplified by Canada [56], to provide a foundation for a better understanding of the variation of tree growth and forest productivity that always become central to sustainable forest management.

Annual tree rings of Sungkai were drawn in this study. Since its annual tree rings are seen clearly, the digitization process easily extracted every point and plotted it in a coordinate system. This digitization is an important starting point for image processing and pattern recognition. Figure 1 shows the data point plots of each tree ring with its actual photograph background. The (x,y) rectangular (Cartesian) transformation into the (r,θ) polar coordinate system enabled the curve-fitting procedure using non-linear regression based on Equation (5) model [17]. The non-linear regression procedure resulted in well-fit ellipse curves with high coefficient determination (0.7842 < R² < 0.9971). This good goodness of fit was also proven by the MSE values (0.0003 < MSE < 3.2269), which were much smaller than the total mean square (

Table 1. The estimated parameters (r_o, a, b, k, θ_o), coefficient of determination (R²), and mean square error (MSE) of each annual tree ring’s ellipse curve-fitting using non-linear regression proposed by Denih et al. [17].

Ring No.	Estimated Parameters					R2	MSE
	ro	a	b	k	θo
1	0.0000	0.8765	1.1781	0.8025	3.5613	0.9729	0.0003
2	0.1163	1.5042	1.9482	0.8072	3.9473	0.7842	0.0095
3	0.2513	2.9366	3.1297	0.7047	5.4538	0.8671	0.0061
4	0.2489	3.6343	3.9814	0.8279	1.1021	0.9015	0.0054
5	0.8042	4.6748	5.1761	1.0099	1.3466	0.9561	0.0162
6	1.4426	6.2761	6.5727	1.0820	1.5254	0.9953	0.0048
7	1.7723	7.0825	7.5684	1.1368	1.5418	0.9919	0.0129
8	1.9608	7.5812	8.4940	1.1923	1.6361	0.9899	0.0195
9	2.4809	8.4677	10.1509	1.1649	1.7225	0.9876	0.0396
10	2.8687	9.7797	11.7342	1.2180	1.7620	0.9907	0.0397
11	3.1853	11.0242	13.4325	1.2311	1.7460	0.9903	0.0561
12	3.6716	12.0681	14.4468	1.2314	1.6617	0.9757	0.1769
13	3.8596	13.6457	16.2887	1.2867	1.5484	0.9744	0.2009
14	5.0054	15.9945	20.2612	1.1999	1.5989	0.9967	0.0465
15	5.2019	18.6072	21.9800	1.2374	1.5929	0.9971	0.0385
16	5.9581	23.8215	25.9090	1.3944	1.6633	0.9870	0.2241
17	6.3170	27.9713	31.9567	1.5382	1.7393	0.9755	0.5365
18	7.0462	32.5126	42.1244	1.5672	1.9532	0.9665	1.3202
19	7.6405	40.5569	50.6191	1.5544	2.1527	0.9455	2.3841
20	6.6908	47.2447	60.3924	1.5659	2.0851	0.9510	2.3775
21	5.9951	51.8527	67.6943	1.5756	2.0804	0.9559	2.3363
22	6.0045	56.2458	73.4963	1.5902	1.8718	0.9477	3.2269
23	5.7924	62.3628	77.9605	1.5799	1.8607	0.9391	3.1598
24	4.9514	67.8478	83.8636	1.5717	1.8981	0.9391	2.8832

).

Table 1 summarizes the estimated parameters of the best-fit ellipse curve in a polar coordinate system for every annual tree ring. The (r_o, θ_o) coordinate indicates the ring center, a and b are the semi-major and semi-minor, and kπ is the rotation angle about the ellipse center. The semi-major and semi-minor values’ rise from pith to outside shows that the diameter and cross-sectional area grow bigger and larger. The next year ring is always outside the previous one, and they do not intersect because the cambium, located between the phloem and xylem, produces the new xylem layer at the outermost wood.

If influenced by genetics only, the cambium differentiation rate of a healthy tree is similar along the stem circumference and generates the same width of the xylem layer; therefore, the stem’s cross-section forms a circle [17]. Figure 1 shows that the cross-section shape is more like an ellipse than a circle, and the distance between two consecutive rings along the circumferential position is not constant. This phenomenon is attributed to the tree’s physiological response to external factors (i.e., environmental distraction, biogeophysical factor, succession, and competition for light, space, and nutrition) during its growth period. The distance variation between two consecutive rings along their circumferential location indicates the reaction wood formation.

Reaction wood forms a wider distance between annual tree rings; it deviates the tree-ring shape from the idealized geometric form and translocates its center from the pith (Figure 2). Consecutive annual tree-ring center translocations may estimate the reaction wood formations, where the angles and distances relate to their positions and width. Equation (20) estimates the reaction wood’s position angles (θ_rw) in a polar diagram system, and

Table 2. The reaction wood formation position, estimated by the tree-ring center’s translocations.

Ring No.	Tree-Ring Center Position		Reaction Wood Formation
	xo	yo	θrw	Start		End
				xrw	yrw	xrw	yrw
1	0.00	0.00	-	-	-	-	-
2	−0.08	−0.08	3.95	−0.81	−0.84	−1.42	−1.48
3	0.17	−0.19	5.90	1.37	−0.56	2.98	−1.21
4	0.11	0.22	1.71	−0.38	2.73	−0.56	3.98
5	0.18	0.78	1.45	0.49	4.11	0.70	5.92
6	0.07	1.44	1.74	−1.00	5.80	−1.36	7.86
7	0.05	1.77	1.61	−0.34	7.99	−0.39	9.26
8	−0.13	1.96	2.34	−5.96	6.16	−6.81	7.03
9	−0.37	2.45	2.03	−4.58	9.19	−5.57	11.17
10	−0.55	2.82	2.01	−5.30	11.33	−6.11	13.05
11	−0.56	3.14	1.60	−0.42	13.61	−0.47	15.31
12	−0.33	3.66	1.17	5.38	12.60	6.02	14.11
13	0.09	3.86	0.45	12.09	5.82	14.32	6.89
14	−0.14	5.00	1.77	−3.68	18.54	−4.74	23.88
15	−0.12	5.20	1.44	2.89	22.44	3.17	24.60
16	−0.55	5.93	2.11	−13.45	22.63	−15.55	26.16
17	−1.06	6.23	2.62	−25.01	14.48	−29.93	17.33
18	−2.63	6.54	2.95	−32.94	6.50	−43.77	8.63
19	−4.20	6.38	3.24	−43.31	−4.25	−53.41	−5.25
20	−3.29	5.83	5.73	32.06	−19.71	39.39	−24.21
21	−2.92	5.23	5.27	21.93	−35.38	24.71	−39.87
22	−1.78	5.73	0.41	61.07	26.75	67.66	29.63
23	−1.66	5.55	5.30	28.68	−42.62	32.34	−48.04
24	−1.59	4.69	4.79	4.25	−56.81	4.72	−63.07

summarizes the results. The starting and ending points of reaction wood are determined by inputting the (θ_rw(t)) and (θ_rw(t+1)) in each best-fit non-linear regression formula following Equation (5). The blue line in Figure 1 shows the reaction wood positions estimated from the tree-ring center translocations.

$${\mathsf{\theta }}_{rw}=\left\{\begin{array}{l}\begin{array}{l}\mathrm{atan}\left(\frac{y_{t+1}-y_t}{x_{t+1}-x_t}\right) ;\mathrm{for} 0\le {\mathsf{\theta }}_{rw}\le \frac{\mathsf{\pi }}{2}\\ \mathsf{\pi }+\mathrm{atan}\left(\frac{y_{t+1}-y_t}{x_{t+1}-x_t}\right) ;\mathrm{for} \frac{\mathsf{\pi }}{2}\le {\mathsf{\theta }}_{rw}\le \mathsf{\pi }\end{array}\\ \begin{array}{l}\mathsf{\pi }+\mathrm{atan}\left(\frac{y_{t+1}-y_t}{x_{t+1}-x_t}\right) ;\mathrm{for} \mathsf{\pi }\le {\mathsf{\theta }}_{rw}\le \frac{3\mathsf{\pi }}{2}\\ 2\mathsf{\pi }+\mathrm{atan}\left(\frac{y_{t+1}-y_t}{x_{t+1}-x_t}\right) ;\mathrm{for} \frac{3\mathsf{\pi }}{2}\le {\mathsf{\theta }}_{rw}\le 2\mathsf{\pi }\end{array}\end{array}\right.$$

(20)

where: $\left(x_t=r_{ot}\cos \left({\mathsf{\theta }}_{ot}\right),y_t=r_{ot}\sin \left({\mathsf{\theta }}_{ot}\right)\right)$ is the coordinate of the tth tree-ring center and $\left(x_{t+1}=r_{o\left(t+1\right)}\cos \left({\mathsf{\theta }}_{o\left(t+1\right)}\right), y_{t+1}=r_{o\left(t+1\right)}\sin \left({\mathsf{\theta }}_{o\left(t+1\right)}\right)\right)$ is the coordinate of the (t + 1)th tree-ring center.

3.2. Growth Curve

A growth curve is commonly divided into four stages, namely exponential, linear, logarithmic, and asymptotic, related to the accelerated rate, constant velocity, decelerated rate, and no significant additional increment, respectively. Juvenile (young) trees grow at an accelerated rate; the full-vigor phase (maturity) grows constantly; the senescent phase grows at a decelerated rate; and old trees do not grow significantly. Although the growth curve may be composed by connecting the four curve types to fit each growth stage, some general formulas that are reliable for curve-fitting all growth stage data have been developed. Gompertz [18] proposed a growth curve model, which resembles a logistic function, that has three arbitrary constants which correspond to the asymptote, the time origin, and the rate constant (Equation (12)). The Gompertz model is the second-most frequently employed sigmoid model, after the logistic model [57], to fit time series-related growth data [58] in an enormous amount of literature. Since its skewness is constant, the logistic (sometimes called Verhulst) curve gives the fixed point of inflection suitable for individual organisms or population growth, which manifests inflection about midway between the upper and lower asymptotes [59]. Many biological science articles reported the Gompertz model in curve-fitting human [60], animal [61,62], and plant [63] growth, as well as in relation to bacteria [64], microbes [65,66], algae [67], and cancer and tumors [68,69]. The Gompertz model belongs to the Richards family of three-parameter growth models that relate to the logistic, the negative exponential, and the von Bertalanffy (Equation (14)). The Richards four-parameter sigmoid growth model (Equation (13)) unified and generalized those familiar growth models amongst many parameterizations, re-parameterizations, and special cases [70].

3.2.1. Diameter

A tree stem may be more similar to truncated cone 3D geometry, a synonym to a conical frustum, than a cylinder [71]; it commonly tappers upward, the bottom diameter is the biggest, and it becomes smaller when nearer to the top. As a product of its evolution, trees commonly have butt swell (bottom thickening), where the stem near the base becomes greatly enlarged to support self-weight loads (i.e., stem and canopy weight) and external loads (i.e., wind and snow mound). Tree stem diameter is commonly measured at breast height; it is easy to perform and results in a meaningful value to avoid overestimated volume. Diameter is the average maximum diameter (major axis) and minimum diameter (minor axis). Diameter at breast height (dbh) is widely used in forest measurement to estimate the tree volume, where the basal area is assumed to be circular.

Table 3. Diameter (cm) and area (cm²) of wood inside each annual tree ring.

Ring No.	Diameter (cm)	Area (cm2)		Ring No.	Diameter (cm)	Area (cm2)
		Circle	Ellipse			Circle	Ellipse
1	0.21	0.03	0.03	13	2.99	7.04	6.98
2	0.35	0.09	0.09	14	3.63	10.32	10.18
3	0.61	0.29	0.29	15	4.06	12.94	12.85
4	0.76	0.46	0.45	16	4.97	19.42	19.39
5	0.99	0.76	0.76	17	5.99	28.21	28.08
6	1.28	1.30	1.30	18	7.46	43.75	43.03
7	1.47	1.69	1.68	19	9.12	65.29	64.50
8	1.61	2.03	2.02	20	10.76	90.99	89.64
9	1.86	2.72	2.70	21	11.95	112.24	110.27
10	2.15	3.64	3.61	22	12.97	132.21	129.87
11	2.45	4.70	4.65	23	14.03	154.65	152.74
12	2.65	5.52	5.48	24	15.17	180.77	178.76

summarizes the sum of the semi-major (a) and semi-minor (b) of the elliptical shape of each annual tree ring; it represents the stem diameter of tree of corresponding age. The plot of diameter vs. age in Figure 3 shows a likely sigmoid curve, nearly similar in shape to the letter S. The curve begins with the increasing diameter with velocity acceleration, generating an exponential form, then the linear and logarithmic states follow. Asymptote, the last part of the growth curve with a relatively constant diameter, does not appear, indicating that the tree has not reached its old phase. The diameter of a 24-year-old live Sungkai tree will still be capable of increasing, although the velocity decelerates.

The Gompertz, von Bertalanffy, and Chapman–Richards growth models were successful in curve-fitting the data plot of time series-related diameter with a high coefficient of determination (R² = 0.988–0.9971) and small mean square error (MSE = 0.0764–0.2999) (

Table 4. Best-fit growth curve models for Sungkai diameter following (a) continuous methods (Gompertz, von Bertalanffy, and Chapman–Richards formula) and (b) discrete methods [23].

Model	Estimated Formula	R2	MSE
a. Continuous methods
Gompertz	$N=33.3761\mathit{exp}\left(-13.6016\mathit{exp}\left(-\left({1.0452}^t\right)\right)\right)$	0.9928	0.1819
von Bertalanffy	$N=-0.9058\left(1-\mathit{exp}\left(0.1219\left(t+0.1182\right)\right)\right)$	0.9880	0.2999
Chapman–Richards	$N=15.5998{\left(1+110742079\mathit{exp}\left(-0.8327t\right)\right)}^{\frac{1}{1-5.8519}}$	0.9971	0.0764
b. Discrete methods
Exp. transformation (linear)	$N_{i+1}=N_i+N_i\left(0.2972-0.018N_i\right)\left(t_{i+1}-t_i\right);N_0=0.1585$	0.9873	0.3196
Exp. transformation (quadratic)	$N_{i+1}=N_i+N_i\left(0.3664-0.0606N_i+0.0032{N_i}^2\right)\left(t_{i+1}-t_i\right);N_0=0.2310$	0.9384	1.6231

). The estimated formulas are given in Table 4, and the graphs are shown in Figure 3a. All three formulas were justifiably reliable in drawing the growth curve of Sungkai diameter with more than a 95% confidence level. As expected, the four-parameter Chapman–Richards sigmoid formula was the best-fit model because it has one more parameter than the others. The Gompertz and von Bertalanffy models have three parameters. The Gompertz model resulted in a better fit estimation than the von Bertalanffy.

In addition to continuous transformation, we also performed the exponential transformation proposed by Bahtiar and Darwis [23] to fit the diameter growth using discrete methods. Discrete methods compose the growth function by assuming that the current dimension (N_i) is the function of the previous one (N_i−1), the next dimension (N_i+1) is the function of the current one (N_i), and so on. The exponential transformation of the dependent and independent variables paves the way to perform simple linear or polynomial regression to graph the best-fit estimated sigmoid growth curve. The discrete method satisfyingly resulted in the best-fit curves drawn in Figure 3b with R² ranging from 0.9384 to 0.9873 and MSE ranging from 0.3196 to 1.6231. The estimated parameters are also tabulated in Table 4. The linear transformation gave a better-fit curve than the quadratic. Although discrete methods have less goodness of fit than continuous ones, they provide simpler procedures to be conducted.

3.2.2. Basal Area

Basal area, calculated as the tree’s cross-sectional area measured at breast height (130 cm above ground), is an important variable in forest mensuration. It is further analyzed to represent the stand density, estimate the merchantable volume of log or timber, measure the forest biomass and complexity, and indicate the forest recovery during succession. Forestry professionals have commonly assumed a basal area circular shape for more than 75 years [17]. Although ellipse geometry can better fit the basal area, the difference is shown in Table 3 to be small, where the maximum difference is 1.1%.

The time series plot of age versus basal area of the specimen and the sigmoid growth curve-fittings are presented in Figure 4. The estimated parameters of each best-fit equation, coefficient of determinations, and mean square errors are given in

Table 5. Best-fit growth curve models for Sungkai basal area following (a) continuous methods (Gompertz, von Bertalanffy, and Chapman–Richards formula) and (b) discrete methods [23].

Model	Estimated Formula	R2	MSE
a. Continuous methods
Gompertz	$N=266.5783\mathit{exp}\left(-50.4572\mathit{exp}\left(-\left({1.0656}^t\right)\right)\right)$	0.9882	5.9706
von Bertalanffy	$N=-5.4175\left(1-\mathit{exp}\left(0.2167\left(t+7.4476\right)\right)\right)$	0.9848	49.4273
Chapman–Richards	$N=239.6532{\left(1+2808.7999\mathit{exp}\left(-0.3838t\right)\right)}^{\frac{1}{1-1.8031}}$	0.9985	5.0236
b. Discrete methods
Exp. transformation (linear)	$N_{i+1}=N_i+N_i\left(0.6179+0.003649N_i\right)\left(t_{i+1}-t_i\right);N_0=0.015$	0.9939	20.3879
Exp. transformation (quadratic)	$N_{i+1}=N_i+N_i\left(0.6525-0.008357N_i+0.00003612{N_i}^2\right)\left(t_{i+1}-t_i\right);N_0=0.01942$	0.9910	31.5218

. All models show good goodness of fit, which is justified by their high coefficient of determination (R² = 0.9848–0.9985) and small mean square error (MSE = 5.0236–49.4273). Like the diameter growth, the Chapman–Richards model consistently gave the best fit among other continuous models, and the discrete exponential modification linear model was a better fit than the quadratic model. The von Bertalanffy and exponential modification quadratic approaches indicated that the basal area growth was only in the exponential stage. In contrast, the others (Gompertz, Chapman–Richards, and exponential transformation linear models) found three growth stages: exponential, linear, and logarithmic.

3.2.3. Volume

A tree’s apical meristem differentiation produces primary growth vertically to heighten its trunk and lengthen its branches; at the same time, the vascular cambium forms the secondary growth horizontally to enlarge its diameter. The primary and secondary growth simultaneously increase with the tree’s age; therefore, a tree’s diameter and height usually correlate strongly. An allometric equation estimates a tree’s height by measuring its diameter. An allometric equation also generally predicts the tree trunk or clear-bole timber volumes. Following Krisnawati and Bustomi’s allometric equation [33],

Table 6. Sungkai clear-bole height (m) and log volume (m³) inside annual tree rings predicted using Krisnawati and Bustomi’s allometric equation [33].

Ring No.	Clear-Bole Height (m)	Volume (m3)		Ring No.	Clear-Bole Height (m)	Volume (m3)
		V1	V2			V1	V2
1	1.20	0.00001	0.00001	13	3.63	0.00235	0.00237
2	1.49	0.00003	0.00002	14	3.93	0.00350	0.00356
3	1.88	0.00008	0.00008	15	4.11	0.00442	0.00452
4	2.06	0.00014	0.00013	16	4.47	0.00675	0.00694
5	2.30	0.00023	0.00023	17	4.83	0.00995	0.01030
6	2.56	0.00040	0.00040	18	5.29	0.01570	0.01639
7	2.70	0.00053	0.00052	19	5.74	0.02381	0.02504
8	2.81	0.00064	0.00064	20	6.15	0.03363	0.03557
9	2.98	0.00087	0.00087	21	6.42	0.04183	0.04442
10	3.17	0.00118	0.00118	22	6.64	0.04959	0.05282
11	3.34	0.00154	0.00155	23	6.86	0.05838	0.06235
12	3.45	0.00182	0.00183	24	7.08	0.06866	0.07354

summarizes the Sungkai’s estimated height and volume at each age, and the data plot is shown in Figure 5.

Non-linear regression successfully fit the volume growth curve with a high goodness of fit (R² = 0.9843–0.9986 and MSE = 0.0000007004–0.00000343) (

Table 7. Best-fit growth curve models for Sungkai volume following (a) continuous methods (Gompertz, von Bertalanffy, and Chapman–Richards formula) and (b) discrete methods [23].

Model	Estimated Formula	R2	MSE
a. Continuous methods
Gompertz	$N=0.10063\mathit{exp}\left(-43.9807\mathit{exp}\left(-\left({1.0668}^t\right)\right)\right)$	0.9983	0.000000799
von Bertalanffy	$N=-0.002040\left(1-\mathit{exp}\left(0.2226\left(t-7.7955\right)\right)\right)$	0.9843	0.000007507
Chapman–Richards	$N=0.0937{\left(1+1899.937\mathit{exp}\left(-0.3711t\right)\right)}^{\frac{1}{1-1.7037}}$	0.9986	0.000000700
b. Discrete methods
Exp. transformation (linear)	$N_{i+1}=N_i+N_i\left(0.6601-9.60995N_i\right)\left(t_{i+1}-t_i\right);N_0=3.4016{10}^{-6}$	0.9937	0.00000343
Exp. transformation (quadratic)	$N_{i+1}=N_i+N_i\left(0.6927-21.2747N_i+219.8693{N_i}^2\right)\left(t_{i+1}-t_i\right);N_0=4.372{10}^{-6}$	0.9941	0.00000340

). The Chapman–Richards model usually produced the best goodness of fit because it had the highest R² and smallest MSE. Similar to the previous basal area growth, the Gompertz, Chapman–Richards, and exponential transformation linear models indicated three stages of volume growth, which are exponential, linear, and logarithmic; meanwhile, the von Bertalanffy and exponential modification quadratic approaches only found the exponential stage.

3.3. Increment and Biological Rotation Age

A basic understanding of increment knowledge is extremely important for forest professionals and forestry scientists to assign accurate decisions for sustainable forest management systems. Increment relates to a forest’s yield, and it is affected by species’ internal aspects (genetic and physiological) and external environmental conditions (climatic, edaphic, and biotic). Increment periodical measurement is commonly on a one-year basis, and it is often stated as the current annual increment (CAI). Figure 6 graphs the best-fit growth curve (the Chapman–Richards), where the abscissa (x) is the age in a unit of years; therefore, the CAI is the first derivation of the growth formula, and MAI is the growth formula divided by the corresponding age.

Figure 6 shows that the CAI and MAI are always positive because the growth is irreversible. Since the growth curve is sigmoid, the CAI begins with a small value and then increases with the tree’s age in the juvenile stage until its maturity; the CAI maximum value is reached at the growth curve inflection point where the second derivation of the growth formula is 0 (zero); and then it decreases afterward until nearly reaching an asymptote of 0 (zero) value of increment. The CAI intersects the MAI at the MAI maximum point, and this point is typically justified as the biological optimum rotation age. If the forest management aims to maximize its long-term yield, the tree is proposed to be felled at the biological rotation age; therefore, this age is often stated as the cutting cycle period. For Sungkai, Figure 6 shows that the CAI meets the MAI at the age of 28 years age and the proposed cutting cycle period will be 30 years because it needs approximately 2 years to reach the breast height (130 cm).

4. Conclusions

Forest inventory includes periodic measurement of tree and stand dimensions in a permanent sampling unit to determine their growth and increment to support a sustainable forest management system. A tree’s age is often unknown, especially in natural and citizen-developed forests, and this presents a problem for growth and increment prediction. We propose to conduct annual tree-ring curve-fitting to determine a tree’s age-related dimension to generate a growth curve and increment. Sungkai (Peronema canescens), a luxurious commercial timber chosen as a case study, resulted in a satisfying growth curve following continuous models (Gompertz, Chapman–Richards, and von Bertalanffy) and discrete models (Bahtiar and Darwis exponential modification). The Chapman–Richards model produced the best-fit sigmoid growth curve. The derivation of the growth formula produces the current annual increment (CAI) and mean annual increment (MAI). It resulted in the optimum biological rotation age or a cutting cycle period of 30 years for the Sungkai plantation.