Testing the Geometric Series Hypothesis for the Ascending Leaf Length Sequence per Culm in Six Indocalamus Bamboo Species

The distribution of leaf sizes along a shoot is a fundamental aspect of plant architecture, but a simple quantitative rule for this pattern has not been widely tested. We examined whether the ascending leaf length sequence per culm in six Indocalamus bamboo species follows a geometric series. This hypothesis is particularly plausible for Indocalamus bamboos because their leaves are concentrated near the culm apex, resulting in relatively uniform light exposure across the leaf cohort. However, the culm architecture is not entirely uniform: some leaves arise from smaller, thinner branches, which could naturally generate a geometric size sequence. To test this hypothesis, we measured the lengths of all leaves from 602 culms across the six species. After excluding culms with fewer than three or more than 90 leaves and those with estimated common ratios exceeding 1.2, we retained 577 culms for final analysis, representing 95.8% of the initial sample. For each culm, we fitted a geometric series model to the ascending leaf length sequence using nonlinear least-squares regression. The goodness of fit was evaluated using the mean absolute percentage error (MAPE) between observed and predicted cumulative leaf lengths. The results showed that the geometric series model fitted the data remarkably well, with 75.39% of the 577 culms having a MAPE below 5% and 96.53% below 10%. The estimated common ratios ranged from 1.01 to 1.19, with a mean of 1.05, indicating a modest but consistent geometric increase in leaf length. Significant differences in both the estimated common ratio and the estimated minimum leaf length were found among species, reflecting interspecific variation in growth strategies. These findings provide strong support for the geometric series as a robust descriptor of leaf length organization per culm in bamboos, offering new insights into the architectural structures that constrain the leaf size distribution per culm.