A total of 122 I. decorus culms were sampled in Hongya County, Meishan, Sichuan Province, China (103°27’53’’ E, 29°55’33’’ N, 504.5 m a.s.l.), and a total of 120 I. longiauritus culms were sampled on the Leshan Normal University Campus, Leshan, Sichuan Province, China (103°44’57’’ E, 29°33’54’’ N, 419.6 m a.s.l.) in November 2024. Indocalamus decorus is native to Meishan, whereas I. longiauritus was introduced from the White Horse Experimental Station of Nanjing Forestry University, Jiangsu Province, China to the current sampling site in 2018, where it has naturalized on the campus. All sampled shoots originated from sprouts appearing in spring 2023. Sampling occurred near the end of their second growth season, ensuring that shoots were mature and leaves had fully matured. Sampling sites for I. decorus and I. longiauritus were selected to ensure minimal anthropogenic disturbance and climatic consistency and similarity. Both locations represent naturalized growth: I. decorus in its native habitat and I. longiauritus naturalized since 2018 without subsequent management. The close distance of 48.6 km between sites minimizes climatic differences, while architectural simplicity (low leaf counts and open canopies) facilitated accurate non-destructive trait measurements. Sampling at this time of year ensured full leaf maturation and dry mass stabilization (i.e., completion of leaf expansion (lamina size stabilized) and sclerification, eliminating ontogenetic biases in area/mass relationships). This rendered oven-drying protocols following standardized methods sufficient to achieve constant dry weight, ensuring stable biomass measurements.

The lamina length (L) and width (W) of each leaf of the 242 shoots were measured. According to prior studies (Shi et al., 2021), individual leaf lamina area (A) can be estimated as the product of L and W multiplied by a proportionality coefficient (k, called the Montgomery parameter) called the Montgomery equation, i.e., A = kLW) as proposed by Montgomery (1911). To determine the numerical value of k, 254 I. decorus leaves and 251 I. longiauritus leaves were scanned using a photo scanner (M208, BenQ Corporation, Shanghai, China). The leaf images were transferred to black-and-white images in bitmap (bmp) format, and the Matlab procedure proposed by Su et al. (2019) was used to extract the planar coordinates for each lamina. The “bilat’ function in the R packaged “biogeom” (v.1.4.3) based on R (v4.3.1; R Core Team, 2023) was used to determine A, L, and W for each leaf. For each shoot, all the leaves were subsequently dried in an oven (DHG-9240A, Shanghai Yiheng Scientific Instruments Co., Ltd., Shanghai, China) at 105°C for 30 min and then continuously at 75°C until achieving a constant dry weight. The dried leaves for each shoot were measured to determine dry mass using an electronic balance (ML203; Mettler Toledo Company, Greifensee, Switzerland; measurement accuracy 0.001 g).

Total leaf area per shoot (A _T) was obtained by summing the individual leaf areas per shoot; the sum of individual leaf widths on a shoot is referred to as W _KS; and the maximum individual leaf length on a shoot is referred to as L _KS. The subscript KS is used in honor of Koyama and Smith.

The raw data of the length and width for all leaves on the 242 shoots, those of the length, width and area for the 505 randomly sampled leaves (used to estimate the Montgomery parameter), and those of the total leaf dry mass for each of the 242 shoots of the two bamboo species are assessable from Supplementary Tables S1 − S3 , respectively.

The frequency distributions of the number of leaves, total leaf mass, and total leaf area per shoot of the two bamboo species were modeled using the normal, log-normal, and two-parameter Weibull probability density functions. The Weibull probability density function takes the form (Weibull, 1951):

where δ is the shape parameter, and λ is the scale parameter. The two parameters in Equation 1 were estimated using maximum likelihood estimation (MLE) which was carried out by the “mle2” function in the R package bbmle (v1.0.25.1). The numerical value of the parameter δ can be used to determine whether the distribution is left-skewed, right-skewed or symmetrical, i.e., δ < 3.6 indicates a right-skewed distribution; δ > 3.6 indicates a left-skewed distribution is indicated; and δ = 3.6 indicates a symmetrical distribution (Murthy et al., 2004).

The log-transformed version of the Montgomery equation (Montgomery, 1911; Shi et al., 2021) was fitted to determine the Montgomery parameter:

We tested the validity of the Koyama and Smith (2022) model, which assumes a one-to-one proportional relationship between A _T and L _KS W _KS (which is referred to as the Montgomery-Koyama-Smith equation, denoted as MKSE), i.e.,

where k _KS is the proportionality coefficient of the MKSE. To stabilize the variances of A _T and L _KS W _KS, we also used the log-transform of Equation 3 for carrying out a linear fitting:

Across diverse plant species, two interdependent variables of Y ₁ and Y ₂, such as leaf area and mass, are frequently found to follow a power-law equation (denoted as PLE) (Niklas, 1994):

where β is the normalization constant and α is the scaling exponent (i.e., the rate of change in Y ₂ with respect to Y ₁; Niklas, 1994) Because α = d Y 2 / Y 2 d Y 1 / Y 1 , it follows that (i) α > 1 indicates that increases in Y ₁ do not keep pace with the increases in Y ₂; (ii) α < 1 indicates that increases in Y ₂ do not keep pace with the increases in Y ₁; and (iii) α = 1 indicates an isometric relationship between Y ₁ and Y ₂. Cases (i) and (ii) are referred to as allometric scaling relationships, whereas case (iii) is referred to as an isometric scaling relationship. We examined whether the A _T vs. L _KS W _KS relationship tends to be allometric or isometric. Log-transformation of Equation 5 takes the form:

where y = log(Y ₂), x = log(Y ₁), and γ = log(β) is the log-log y-intercept. Equation (5) was used to describe the probable scaling relationship between A _T and L _KS W _KS. In addition, it was also used to describe the scaling relationships between A _T and the total number of leaves per shoot (N _T), and between A _T and M _T, where M _T represents total leaf dry mass per shoot.

Given that LW has the same physical dimensions as A, and L _KS W _KS has the same dimensions as A _T, common least-squares regression was used to fit Equations 2, 4 and 6. However, both A _T and N _T, and A _T and M _T have different physical dimensions. Thus, reduced major axis protocols (Niklas, 1994; Quinn and Keough, 2002) were used to estimate the slope and y-intercept of the A _T vs. N _T and the A _T vs. M _T scaling relationships. The root-mean-square error (RMSE) and coefficient of determination (r ²) were used to assess the goodness of fit of the linear fitting.

The bootstrap percentile method (Efron and Tibshirani, 1993) was used to calculate the 95% confidence intervals (CIs) of α and β. The bootstrap percentile method based on 3000 bootstrap replicates was used to test the significance of the difference between any two scaling exponents and any two normalization constants between the two bamboo species. If the 95% CI of the differences between the bootstrap replicates of one scaling exponent (or one normalization constant) and those of another scaling exponent (or another normalization constant) included zero, the difference between the two statistical parameters was judged not to be significant; if the 95% CI did not include zero, the difference was judged to be statistically significant.

All analyses were conducted in R (v4.3.1; R Core Team, 2023).

The MKSE posits a strict isometric scaling relationship (α = 1) between A _T and the composite metric L _KS W _KS (i.e., the product of the sum of leaf widths and maximum leaf length). The rationale for assuming constant k _KS in the MKSE merges from the foundational Montgomery equation (individual leaf area A ∝ LW), where a species-specific constant k captures the geometric proportionality between A and the LW product. Extending this proportionality to the shoot level (A _T ∝ L _KS W _KS) implicitly assumes that the average proportionality (k) across all leaves within a shoot remains constant, or that variations cancel out, allowing L _KS W _KS to serve as a dimensionally consistent proxy for total area. Furthermore, prior theoretical work (Meng et al., 2025) shows that, if four strict conditions hold true simultaneously for shoots, A _T scales isometrically with L _KS W _KS (α = 1): (1) the number of leaves is constant across shoots, (2) individual leaf areas sorted in ascending order for each shoot follow a geometric series, (3) the common ratio of this geometric series is constant across shoots, and (4) individual leaf width scales (isometrically or allometrically) with leaf length. However, our analyses reveal a significant sub-linearity (α < 1) in both species ( Figures 4B, D ), indicating that A _T increases at a disproportionately slower rate than L _KS W _KS as shoots increase in overall size. This deviation from the idealized MKSE proportionality arises from intrinsic architectural heterogeneities neglected in the model’s assumptions. The MKSE requires that the numerical value of the Montgomery parameter (k) is constant or nearly so across all leaves per shoot (Koyama and Smith, 2022). However, in the case of Indocalamus, pronounced intra-shoot leaf size variation is evident, which is interpreted to reflect an ontogenetic gradient in leaf development along the length of shoots (e.g., smaller basal leaves vs. larger apical leaves), which results in a basipetal shift in the numerical value of k (Shi et al., 2021). As shoot size increases, larger leaves tend to exhibit lower k-values due to the thickening of vascular bundles and the mesophyll, reducing the area gained per unit LW (Meng et al., 2025). Consequently, L _KS W _KS overestimates A _T in larger shoots, as dimensional increases in L _KS W _KS outpace actual area expansion (A _T). This aligns with the results reported by Meng et al. (2025), who observed that MKSE yields less than optimal results using Sasaella kongosanensis ‘Aureostriatus’, presumably because of inconsistent leaf-number-dependent allometries. In the case of Indocalamus, architectural simplicity (low leaf counts, and open canopies) likely reduces (but does not eliminate) this bias, highlighting the fact that plants tend to optimize leaf deployment not for geometric proportionality but for light capture efficiency as a consequence of hydraulic and mechanical trade-offs (Givnish, 1984; Niklas, 1994; Westoby et al., 2002). Thus, scaling exponents with numerical values less than unity (α < 1) identify the “cost” of producing larger shoots, where diminishing returns in area gain per unit dimensional investment reflect selection for reducing self-shading and enhancing hydraulic safety (Smith et al., 2017; Wang et al., 2024).

Our findings confirm that structural heterogeneity within shoots, particularly the ontogenetic gradient in leaf size and associated variation in the Montgomery parameter (k), is a key driver of the observed deviation from strict proportionality in the MKSE. It is therefore acknowledged that intra-shoot variability in k-values represents a potential source of prediction error, especially in shoots with more complex architecture or pronounced developmental gradients. Future studies should focus on the quantification of k-value variation across different leaf sizes, phyllotaxies, and developmental stages. Such analyses are crucial for assessing the robustness of the MKSE across diverse plant forms and for developing refined predictive models that account for inherent structural heterogeneity.

The nonlinear scaling of A _T vs. L _KS W _KS observed when α < 1 also mirrors light attenuation patterns in tropical canopies. Kitajima et al. (2005) quantified how late-successional trees with orthotropic branching enhance shading through high cumulative leaf area index (LAI) ranging between 5 and 8 despite low light extinction coefficients (K = 0.35), whereas pioneers maintain minimal LAI (< 0.5) for maximizing light penetration. In the case of Indocalamus, similar architectural optimization occurs, e.g., larger shoots exhibit diminishing area gains per unit dimensional increases (L _KS W _KS), reflecting trade-offs between hydraulic safety and light capture efficiency. Although our models demonstrate robustness in the studied populations, environmental factors (e.g., light, water, or nutrient gradients) may influence scaling parameters in heterogeneous habitats, e.g., resource limitation could exacerbate diminishing returns in the scaling of A _T/L _KS W _KS (α < 1) by constraining leaf expansion. Future studies manipulating environmental stressors are needed to test the plasticity of these relationships across ecological settings.

Our data reveal an interspecific contrast in how A _T scales with respect to leaf number (N _T) in two closely related species. Specifically, we observe an allometric A_T vs. N_T scaling relationship in the case of I. decorus (α < 1) and an isometric scaling relationship in the case of I. longiauritus (α ≈ 1) ( Figures 5A, B ). We interpret this difference to reflect adaptations to divergent self-shading intensity, modulated by canopy architecture. Specifically, I. decorus shoots, on average, bear twice as many leaves per shoot (mean N _T = 26.3) compared to I. longiauritus (mean N _T ​= 12.5), and consequently experience significantly more self-shading. Consequently, I. decorus exhibits an allometric scaling relationship (i.e., A T ∝ N T 0.889 ), such that the addition of each leaf contributes disproportionately more to total leaf area per shoot. This observation differs from the data reported by Wang et al. (2024), who observed an A _T vs. N _T scaling relationship governed by α = 1.128 in Shibataea chinensis (experiencing considerable self-shading), compared to α = 0.820 in Sasaella kongosanensis ‘Aureostriatus’ (experiencing little or no self-shading). The difference between α > 1 in S. chinensis and α < 1 in I. decorus (both high self-shading species) likely reflects distinct architectural adaptations to shading. S. chinensis mitigates shading by means of an even vertical leaf distribution, where the addition of leaves necessitates disproportionate area increases (α > 1) to compensate for layered shading. In contrast, I. decorus exhibits apical leaf clustering, where adding leaves expands existing clusters rather than adding new shaded layers. This architecture imposes stronger self-thinning constraints, limiting area gains per added leaf (α < 1) despite high overall shading. Thus, though both species experience considerable self-shading, their divergent leaf deployment strategies drive contrasting scaling exponents. Thus, the numerical value of the A _T vs. N _T scaling exponent emerges as an important functional signature of species-specific light-harvesting optimization as summarized by Corner’s rules (Corner, 1949), which proposes that branching geometry dictates whether area expansion prioritizes leaf size or number (Olson et al., 2018).

Our observations of crown architecture also align with recent discoveries of life-form-dependent shade avoidance strategies. For example, Aoyagi et al. (2024) have shown that Japanese monoaxial trees exhibit divergent leaf arrangements, i.e., deciduous species minimize self-shading through top-down (basipetal) decreasing petiole lengths (r = −0.94) and deflection angles, enabling apical leaf expansion, whereas evergreen species employ bottom-up (acropetal) increasing angles (r = 0.82) to create vertical inter-whorl spacing. This dichotomy underscores how phylogenetic constraints shape crown plasticity, which is evident in I. decorus (high leaf count) and I. longiauritus (low leaf count), where architectural simplicity permits flexible adaptations to self-shading.

We suspect that the A_T vs. M_T allometry integrates two key trade-offs: (i) potentially competing leaf-level structural and physiological requirements, and (ii) shoot-level leaf biomass partitioning trade-offs. Across diverse species, individual leaves tend to exhibit a positive M vs. A scaling allometry ( M ∝ A α > 1 ), wherein larger leaves require disproportionately more mass with increasing lamina area, presumably for mechanical and hydraulic tissue systems (Milla and Reich, 2007; Niklas et al., 2007; Niinemets et al., 2007). However, in the case of Indocalamus, larger shoots appear to produce thinner leaves or leaves with reduced tissue density to optimize carbon allocation. In Indocalamus, both mechanisms likely operate. First, the conserved Montgomery parameter (k ≈ 0.72) indicates geometric similarity in leaf shape, but the shoot-level A _T vs. M _T allometry (α = 0.912 and 0.936 for the two investigated bamboo species) indicates increasing mass per unit area. This agrees with the “diminishing returns” pattern observed in subtropical ferns, where total leaf area scales allometrically with biomass (α < 1.0) across elevations (Chen et al., 2023). This further indicates that there are similar resource allocation strategies governing the A _T vs. M _T scaling relationship, i.e., increased individual plant (or shoot) size requires increases in LMA to enhance structural support under variable light regimes. Second, as shoots enlarge, resource allocation shifts toward higher LMA. This increased investment per unit area supports greater structural integrity (e.g., thicker mesophyll, denser vascular bundles) and potentially enhanced longevity or defense, which may be advantageous in environments where larger shoots experience greater mechanical stress or herbivory pressure, even within generally resource-rich habitats (Wright et al., 2004; Poorter et al., 2009). Importantly, the A _T vs. M _T allometry (α < 1) confirms the “diminishing returns” pattern reported for whole-plant LMA scaling (Niklas et al., 2007). This reflects an economic strategy where larger shoots exhibit a relative increase in biomass investment per unit photosynthetic area (higher LMA). While this prioritization of structural biomass over maximal area expansion (A _T/M _T decreases as M _T increases) might seem to constrain potential photosynthetic gains per unit carbon invested, it likely enhances hydraulic safety, mechanical stability, and leaf longevity in larger architectural units. This strategy optimizes carbon gain under the specific constraints faced by larger shoots (e.g., greater self-loading, longer hydraulic pathways), rather than maximizing area per se under non-limiting conditions (Poorter et al., 2009). The conserved low average LMA characteristic of fast-growing bamboos like Indocalamus (Wright et al., 2004) is thus achieved alongside this ontogenetic increase in LMA within shoots as they develop.