The quadtree data structure is commonly used in image coding to decompose an image into separate spatial regions to adaptively identify the type of quantizer used in various regions of an image. The authors describe the theory needed to construct quadtree data structures that optimally allocate rate, given a set of quantizers. A Lagrange multiplier method finds these optimal rate allocations with no monotonicity restrictions. They use the theory to derive a new quadtree construction method that uses a stepwise search to find the overall optimal quadtree structure. The search can be driven with either actual measured quantizer performance or ensemble average predicted performance. They apply this theory to the design of a motion compensated interframe video coding system using a quadtree with vector quantization.