Does a DNA sequence has its own derivation tree or pattern?
From your example I guess that by "deriving" you mean something like decomposing into elements with diverse functions. Maybe we could apply this concept to DNA sequences: Some parts code proteins, which are decomposed into triplets, each of which codes an amino-acid. Other parts have regulatory functions: They can be motifs not coding a protein, but instead allowing a protein or something else to bind and modify the expression of nearby protein-coding parts. Other parts may contribute to the physical structure of the chromosome, or may have no function at all.