In this paper, we first introduce especial subsets of a bounded lattice, called bottom (resp. top) branches, and suitable associative operations on them, called branch (resp. dual branch) operations, which handily generalize t-subnorms (resp. t-superconorms). Then we propose a more general method for constructing uninorms in U with a given underlying t-conorm (resp. t-norm) and a given underlying branch operation (resp. dual branch operation). We also provide an example to show that not all uninorms in U are representable as in formula given by that more general construction method. In addition, we introduce the subclass of the class U of uninorms on a bounded lattice and lay bare the structure of their members. It is shown that uninorms in U are characterized by a t-conorm (resp. t-norm) and a branch operation (resp. dual branch operation).