Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees

Some of the most interesting de Bruijn sequences can be constructed in seemingly unrelated ways. In particular, the "Granddaddy" and "Grandmama" can be understood by joining necklace cycles into a tree using simple parent rules, or by concatenating smaller strings (e.g., Lyndon words) in lexicographic orders. These constructions are elegant, but their equivalences seem to come out of thin air, and the community has had limited success in finding others of the same ilk. We aim to demystify the connection between cycle-joining trees and concatenation schemes by introducing "concatenation trees". These structures combine binary trees and ordered trees, and traversals yield concatenation schemes for their sequences. In this work, we focus on the four simplest cycle-joining trees using the pure cycling register (PCR): "Granddaddy" (PCR1), "Grandmama" (PCR2), "Granny" (PCR3), and "Grandpa" (PCR4). In particular, we formally prove a previously observed correspondence for PCR3 and we unravel the mystery behind PCR4. More broadly, this work lays the foundation for translating cycle-joining trees to known concatenation constructions for a variety of underlying feedback functions including the complementing cycling register (CCR), pure summing register (PSR), complementing summing register (CSR), and pure run-length register (PRR)