Fig. 5: Illustration of the laziness of branching morphogenesis utilizing the drainage areas of channel tips.

a An example from Saeftinghe, Netherlands, depicting the spatial distribution of drainage areas at all channel tips across a drainage platform (i.e., \({A}_{{tip}}\)). The colored blocks denote the catchment of each tidal network, with the gray lines representing channel lines. The white patches with thin gray borders indicate the drainage areas of the channel tips. b Frequency distribution of \({A}_{{tip}}^{*}\) (\(={A}_{{tip}}{D}_{d}^{2}\)) calculated from the Saeftinghe drainage platform, with the median value of \({A}_{{tip}}^{*}\) also noted. c Variation in the space-filling deviation index \({D}_{i}\) with the median of \({A}_{{tip}}^{*}\). The gray points denote the field data collected in this study. The small, gradient-colored dots indicate the results of more than 6000 model simulations with varying \({A}_{{act}}\) values substituted into Eq. (3). The pink dashed lines delineate the characteristic thresholds derived from \({T}_{{square}}\). d Detailed view of the dashed box in the diagram of (c). The light gray squares correspond to the model results (consistent with the gradient-colored dots in (c)). The circles of varying sizes and colors denote the field data, detailed with mean tidal range and vegetation coverage information. The green line indicates the linear regression of the field data from vegetated flats, with a transparent band showing 95% confidence intervals.