The Geometric Series Hypothesis of Leaf Area Distribution and Its Link to the Calculation of the Total Leaf Area per Shoot of Sasaella kongosanensis 'Aureostriatus'

Total leaf area per shoot (A_T) can reflect the photosynthetic capacity of a shoot. A prior study hypothesized that A_T is proportional to the product of the sum of the individual leaf widths per shoot (L_KS) and the maximum individual leaf length per shoot (W_KS), referred to as the Montgomery-Koyama-Smith equation (MKSE). However, empirical evidence does not support such a proportional relationship hypothesis, as A_T was found to allometrically scale with L_KSW_KS, i.e., AT∝LKSWKSα, where α≠1, referred to as the power law equation (PLE). Given that there is variation in the total number of leaves per shoot (n), little is known about whether the leaf area distribution has an explicit mathematical link with the sorted leaf area sequence per shoot, and it is unknown whether the mathematical link can affect the prediction accuracy of the MKSE and PLE. In the present study, the leaves of 500 shoots of a dwarf bamboo (Sasaella kongosanensis 'Aureostriatus') were scanned, and the leaf area, length, and width values were obtained by digitizing the leaf images. We selected the shoots with n ranging from 3 to 10, which accounted for 76.6% of the totally sampled shoots (388 out of 500 shoots). We used the formula for the sum of the first j terms (j ranging from 1 to n) of a geometric series (GS), with the mean of the quotients of any adjacent two terms (denoted as q¯A) per shoot as the common ratio of the GS, to fit the cumulative leaf area observations. Mean absolute percentage error (MAPE) was used to measure the goodness of fit of the GS. We found that there were 367 out of 388 shoots (94.6%) where 1 < q¯A < 1.618 and MAPE < 15%, and these 367 shoots were defined as valid samples. The GS hypothesis for leaf area distribution was supported by the result that the MAPE values for most valid samples (349 out of 367, i.e., 95.1%) were smaller than 5%. Here, we provide a theoretical basis using the GS hypothesis to demonstrate the validity of the MKSE and PLE. The MAPE values for the two equations to predict A_T were smaller than 5%. This work demonstrates that the leaf area sequence per shoot follows a GS and provides a useful tool for the calculation of total leaf area per shoot, which is helpful to assess the photosynthetic capacity of plants.