Rooted tree graphs and de Bruijn graphs

The de Bruijn graph G_n illustrates all possible transitions in an n-stage binary shift register. Most feedback functions for an n-stage shift register partition the state space into cycles whose combined lengths total 2ⁿ. These cycles are equivalent to factors in the de Bruijn graph. Short cycles from two special feedback functions can be merged to create two subsets of the de Bruijn sequences. Rooted tree graphs count the number of sequences in these two subsets. This methodology can be generalized to construct most de Bruijn sequences and to count the sizes of these other subsets.