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The Max-Tree designed for morphological attribute filtering in image processing, is a data structure in which the nodes represent connected components for all threshold levels in a data set. Attribute filters compute some attribute describing the shape or size of each connected component and then decide which components to keep or to discard. In this paper, we augment the basic Max-Tree data structure such that interactive volumetric filtering and visualization becomes possible. We introduce extensions that allow (1) direct, splatting-based, volume rendering; (2) representation of the Max-Tree on graphics hardware; and (3) fast active cell selection for isosurface generation. In all three cases, we can use the Max-Tree representation for visualization directly, without needing to reconstruct the volumetric data explicitly. We show that both filtering and visualization can be performed at interactive frame rates, ranging between 2.4 and 32 frames per seconds. In contrast, a standard texture-based volume visualization method manages only between 0.5 and 1.8 frames per second. For isovalue browsing, the experimental results show that the performance is comparable to the performance of an interval tree, where our method has the advantage that both filter threshold browsing and isolevel browsing are fast. It is shown that the methods using graphics hardware can be extended to other connected filters.