In savannas, predicting how vegetation varies is a longstanding challenge. Spatial patterning in vegetation may structure that variability, mediated by spatial interactions, including competition and facilitation. Here, we use unique high-resolution, spatially extensive data of tree distributions in an African savanna, derived from airborne Light Detection and Ranging (LiDAR), to examine tree-clustering patterns. We show that tree cluster sizes were governed by power laws over two to three orders of magnitude in spatial scale and that the parameters on their distributions were invariant with respect to underlying environment. Concluding that some universal process governs spatial patterns in tree distributions may be premature. However, we can say that, although the tree layer may look unpredictable locally, at scales relevant to prediction in, e.g., global vegetation models, vegetation is instead strongly structured by regular statistical distributions.