We present the first systematic treatment of the problems posed by the visualization and analysis of large-scale, parallel tree-based adaptive mesh refinement (AMR) simulations on an Eulerian grid. When compared to those obtained by constructing an intermediate unstructured mesh with fully described connectivity, our primary results indicate a gain of at least 80% in terms of memory footprint, with a better rendering while retaining similar execution speed. We thus describe herein the key concepts that allowed us to obtain these results, together with the methodology that facilitates the design, implementation, and optimization of algorithms operating directly on such refined meshes. In 2019, this native support for AMR meshes has been contributed to the open source Visualization Toolkit (VTK). We note that this work pertains to a broader long-term vision, with the dual goal to both improve interactivity when exploring such data sets in 2 and 3 dimensions, and optimize resource utilization.