Data from: A lognormal distribution of the lengths of terminal twigs on self-similar branches of elm trees

Lognormal distributions and self-similarity are characteristics associated with a wide range of biological systems. The sequential breakage model has established a link between lognormal distributions and self-similarity and has been used to explain species abundance distributions. To date, however, there has been no similar evidence in studies of multicellular organismal forms. We tested the hypotheses that the distribution of the lengths of terminal stems of Japanese elm trees (Ulmus davidiana), the end products of a self-similar branching process, approaches a lognormal distribution. We measured the length of the stem segments of three elm branches and obtained the following results: (1) each occurrence of branching caused variations or errors in the lengths of the child stems relative to their parent stems; (2) the branches showed statistical self-similarity; the observed error distributions were similar at all scales within each branch; and (3) the multiplicative effect of these errors generated variations of the lengths of terminal twigs that were well approximated by a lognormal distribution, although some statistically significant deviations from strict lognormality were observed for one branch. Our results provide the first empirical evidence that statistical self-similarity of an organismal form generates a lognormal distribution of organ sizes.

对数正态分布（lognormal distribution）与自相似性，是广泛存在于各类生物系统中的典型特征。连续断裂模型（sequential breakage model）已证实了对数正态分布与自相似性之间的内在关联，并被用于解释物种多度分布。然而迄今为止，针对多细胞生物形态的相关研究中，尚未有类似的实证依据。我们针对以下假说开展了验证：作为自相似分枝过程的最终产物，日本榆（Ulmus davidiana）的末端茎干长度分布趋近于对数正态分布。我们对3根榆树枝的茎段长度进行了测量，得到如下结果：(1) 每一次分枝事件均会导致子茎相对于母茎的长度出现偏差或误差；(2) 受试枝条呈现出统计自相似性（statistical self-similarity），单根枝条在所有尺度下观测到的误差分布均保持一致；(3) 上述误差的累积倍增效应，使得末端细枝的长度变异呈现出良好的对数正态分布拟合效果，尽管其中一根枝条的分布与严格的对数正态分布存在统计学意义上的显著偏离。本研究首次通过实证数据证实：生物形态的统计自相似性，可催生器官尺寸的对数正态分布。