I assume you mean that you want to represent a set of binary strings with a graph or hypergraph (since you mention a binary tree which I think is a set of binary strings - a unique string for every root-to-leaf path, like a Huffman tree.)
The whole field of automata theory explores various graph (machine) models that can represent (i.e. compute, i.e. decide which are and are not in a set of) strings. Pick up any textbook on computability theory and it will present Deterministic Finite Automata (DFAs), Non-deterministic ones (NFAs), variants of those, push-down automata (deterministic and non-deterministic), and Turing machines (which also have a graph representation).
A deBruijn graph is a special case of a DFA, one with a very specialized structure.