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One can use recursive properties better if the structure divides equally. So if not having a strict quadtree is okay, they can divide the square into 3x3, 5x5, etc

For odd size squares, simply duplicate the center row and column before rotating, then deduplicate after.

Another approach would be to allow a cell's value to be null, meaning it should be clipped, but then you run the risk of having transforms on the region quadtrees that cause the null cells to migrate inwards into the square (such that they cannot simply be clipped off later).