We investigate the relationship between two kinds of vertex colorings of
hypergraphs: unique-maximum colorings and conflict-free colorings. In a
unique-maximum coloring, the colors are ordered, and in every hyperedge of the
hypergraph the maximum color appears only once. In a conflict-free coloring, in
every hyperedge of the hypergraph there is a color that appears only once. We
concentrate in hypergraphs that are induced by paths in tree graphs.