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Connections in image processing are an important notion that describes how pixels can be grouped together according to their spatial relationships and/or their gray-level values. In recent years, several works were devoted to the development of new theories of connections among which hyperconnection (h-connection) is a very promising notion. This paper addresses two major issues of this theory. First, we propose a new axiomatic that ensures that every h-connection generates decompositions that are consistent for image processing and, more precisely, for the design of h-connected filters. Second, we develop a general framework to represent the decomposition of an image into h-connections as a tree that corresponds to the generalization of the connected component tree. Such trees are indeed an efficient and intuitive way to design attribute filters or to perform detection tasks based on qualitative or quantitative attributes. These theoretical developments are applied to a particular fuzzy h-connection, and we test this new framework on several classical applications in image processing, i.e., segmentation, connected filtering, and document image binarization. The experiments confirm the suitability of the proposed approach: It is robust to noise, and it provides an efficient framework to design selective filters.