Rooted trees and isometry classes for a ternary linear code

I've just recently had a look at rooted trees and wrote up a short
pedagogical note on the subject at http://tcjpn.spaces.live.com/
under "A Walk in the Woods with Cayley and Comtet". In item 3) of the
note on page 7, I surmise (from some brute force calculations) a
relation between certain forests of rooted trees and the isometry
classes for a ternary linear code (A034363 in the OEIS). I'd be
interested in any comments on the validity of the conjecture or deeper
insights on the connections. (Any constructive criticism of the note
as a whole would be welcome also.)

Sincerely,

TC

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