## Data source

The data used in the present study were related to 1-year-old (April 1985 to March 1986) and 2-year-old (October 1981 to October 1982) seedlings of hinoki cypress (*Chamaecyparis obtusa* [Sieb. et Zucc.] Endl.) growing at Midorigaoka Nursery, Gifu District Forest Office, Minokamo, Gifu Prefecture, central Japan (cf. Ogawa 1989). The planted seedling density was 60 seedlings m− 2 for 1-year-old seedlings and 62 seedlings m− 2 for 2-year-old seedlings.

The nursery is situated at an elevation of 84 m above sea level. The meteorological records from the nursery show that the mean annual precipitation, annual air temperature, warmth index, and coldness index values over the five years between 1981 and 1985 were 1,785 mm yr− 1, 14.1°C, 116.1°C month, and − 7.1°C month, respectively.

## Seedling size and mass measurement

Sampling of seedlings was conducted monthly from April 1985 to March 1986 for 1-year-old seedlings, and at semi-monthly intervals from October 1981 to October 1982 for 2-year-old seedlings. In total, 25 individuals were harvested at each sampling time from the seedbed, totaling 300 1-year-old seedlings and 618 2-year-old seedlings.

After harvesting of the sample seedlings, the seedling height, crown base height, stem diameter at ground level, stem diameter at one-tenth of the height, and stem diameter at the crown base were measured (Table 1).

At each harvest, the seedlings were divided into leaves, roots, and stems. All parts of the sampled seedlings were oven-dried at 85°C for 24 h, transferred to desiccators, and weighed after cooling.

## Stratified clipping method

The stratified clipping method (Monsi and Saeki 1953) was applied to 2-year-old seedlings from October 28, 1981, through October 18, 1982, on a total of 24 occasions. Stratified clipping was conducted at monthly intervals in October 1981 and 1982 and at semi-monthly intervals from November 1981 to September 1982. Overall, 10 seedlings were subjected to stratified clipping at each sampling time from October 1981 to September 1982, and 15 seedlings were harvested in October 1982, totaling 256 seedlings.

At each clipping time, the above-ground parts of the sampled seedlings were divided into leaves, stems, and branches in each 4-cm (from October 1981 to June 1982) or 6-cm (from July 1982 to October 1982) horizontal layer. The sorted plant parts collected in each horizontal layer were oven-dried at 85°C for 24 h, transferred to desiccators, and weighed after cooling.

## Model descriptions

*Relationships of the cumulative leaf mass [F(z)] with the density of non-photosynthetic organs [C(z)] and sapwood area [S* *S* *(z)]*

The pipe-model theory proposed by Shinozaki et al. (1964a, b) empirically demonstrated that the *C*(*z*) (mass tree–1 length–1; i.e., stems and branches) per tree in terms of biomass above the level at a certain distance (length) from the top of the crown [*z*] is proportional to the *F*(*z*) (mass tree–1) above the *z* horizon.

In the present model, the relationship between *F*(*z*) and *C*(*z*) is fitted to a non-rectangular hyperbola, which is often used to describe the rate of single-leaf gross photosynthesis (Thornley 1976; Johnson and Thornley 1984). The non-rectangular hyperbola is expressed as the lower root of the quadratic equation:

$$\theta {F\left(z\right)}^{2}-\left(\alpha C\left(z\right)+{F}_{max}\right)F\left(z\right)+\alpha C\left(z\right){F}_{max}=0$$

1

for which:

$$F\left(z\right)=\frac{1}{2\theta }\left(\alpha C\left(z\right)+{F}_{max}-\sqrt{{\left(\alpha C\left(z\right)+{F}_{max}\right)}^{2}-4\theta \alpha C\left(z\right){F}_{max}}\right)$$

2

where α is the initial slope of the curve (cm), *F*max is the limiting value of *F*(*z*) at the saturation limit of *C*(*z*) (g seedling− 1), and θ is a dimensionless parameter indicating the convexity of the curve (\(0\le \theta \le 1)\). Ogawa (2015) successfully described the *F*(*z*)–*C*(*z*) relationship given by Eq. (4) for adult trees of several different species by using Eq. (2). In this study, whether Eq. (2) is also applicable to tree seedlings will be examined, including another case such as Eqs. (3) as well as Eq. (4).

For θ = 0, Eq. (1) reduces to the rectangular hyperbola:

## \(F\left(z\right)=\frac{\alpha C\left(z\right){F}_{max}}{\alpha C\left(z\right)+{F}_{max}}\) =\(\alpha \left(1-\frac{F\left(z\right)}{{F}_{max}}\right)C\left(z\right)\) (3)

In the special case in which the curve of the non-rectangular hyperbola (Eq. 2) does not converge to fit the data, Eq. (2) for θ = 0 or the rectangular hyperbola (Eq. 3) was fitted rather than the non-rectangular hyperbola for \(0<\theta <1\).

When θ = 1, the following limiting response is obtained:

$$F\left(z\right)=\left\{\begin{array}{c}\alpha C\left(z\right), 0\le C\left(z\right) \le \frac{{F}_{max}}{\alpha }\\ {F}_{max}, C\left(z\right) >\frac{{F}_{max}}{\alpha }\end{array}\right.$$

4

Intermediate values of θ generate response curves lying between these two extremes. The extreme case of Eq. (4) corresponds to the basic concept of the pipe-model theory proposed by Shinozaki et al. (1963a), with α and *F*max defined as the specific pipe length and the total leaf mass of a tree, respectively.

According to Ogawa (2015), the *C*(*z*) at a given depth *z* is expressed as follows:

$$C\left(z\right)=\sigma \left(z\right)S\left(z\right)$$

5

where *S*(*z*) and *σ*(*z*) are the cross-sectional area of the non-photosynthetic organs and the wood density at depth *z*, respectively.

If the proportion of bark area at depth *z* is assumed to be negligible and the sapwood area and the proportional heartwood area are represented by *S*S(*z*) and η(*z*), respectively, *S*(*z*) is determined as:

$$S\left(z\right)=\frac{1}{1-\eta \left(z\right)}{S}_{S}\left(z\right)$$

6

After combining Eqs. (5) and (6), Eq. (4) can be rewritten as:

$$F\left(z\right)=\alpha \sigma \left(z\right)\frac{1}{1-\eta \left(z\right)}{S}_{S}\left(z\right)$$

7

Eq. (7) indicates that *F*(*z*) is proportional to *S*S(*z*), as in Eq. (4), if η(*z*) is constant, whereas *F*(*z*) is not proportional to *S*S(*z*) if η(*z*) is not constant.

*Allometric relationship between leaf mass (m* *L* *) and the square of stem diameter at the crown base (D* *B* *2* *)*

As *S*(*z*) is the cross-sectional area of the non-photosynthetic organs at depth *z*, *S*(*z*) = *S*(*z**) at a depth of *z**\(\)from the top of the crown (i.e., the crown base height), which can be expressed as follows:

$$S\left({z}^{*}\right)=\frac{\pi }{4}{D}_{B}^{2}$$

8

where *D*B is the stem diameter at the crown base.

Considering Eqs. (5) and (8), the following relationship between tree *m*L \(\left(=F\left({z}^{*}\right)\right)\) and *D*B2 can be derived from the two extremes of Eq. (3) for θ = 0 and Eq. (4) for θ = 1:

$${m}_{L}=\frac{\pi }{4}\alpha \left(1-\frac{{m}_{L}}{{F}_{max}}\right)\sigma \left({z}^{*}\right){D}_{B}^{2}$$

9

$${m}_{L}=\frac{\pi }{4}\alpha \sigma \left({z}^{*}\right){D}_{B}^{2}$$

10

where α in Eq. (10) is equal to the specific pipe length (*L*) because the *F*(*z*)–*C*(*z*) relationship described by Eq. (4) is linear. At intermediate values of θ, namely 0 < θ < 1, Eq. (2) is rewritten as the following *m*L–*D*B2 relation:

$${m}_{L}=\frac{1}{2\theta }\left(\frac{\pi }{4}\alpha \sigma \left({z}^{*}\right){D}_{B}^{2}+{F}_{max}-\sqrt{{\left(\frac{\pi }{4}\alpha \sigma \left({z}^{*}\right){D}_{B}^{2}+{F}_{max}\right)}^{2}-\pi {F}_{max}\alpha \sigma \left({z}^{*}\right){D}_{B}^{2}}\right)$$

11

If the terms \(\alpha \left(1-\frac{{m}_{L}}{{F}_{max}}\right)\sigma \left({z}^{*}\right)\) in Eq. (9) and \(\alpha \sigma \left({z}^{*}\right)\) in Eq. (10) are constant, *m*L scales as *D*B2. In contrast, in the range of θ between 0 and 1, the allometric scaling relation between *m*L and *D*B2 is not represented by Eq. (11).

## Regression analysis

The bivariate relationship, of *m*L–*D*B2 (Eq. 9 or 10) was analyzed with standardized major axis (SMA) regression (Warton et al. 2006) and ordinary least squares (OLS) regression using the smatr package of R (v. 4.1.2, R Core Development Team, 2021). Significant differences among power (i.e., scaling) exponents were based on 95% confidence intervals (CIs).

Fitting of the nonlinear equation (Eq. 2) to the data was performed using KaleidaGraph software (v. 5.0.2, Synergy Software, Reading, PA), which is based on the Levenberg-Marquardt algorithm (Press et al. 1992), and the coefficient of determination (*R*2) was used to test for goodness of fit.