  The theory of multichannel prefix codes aims to generalize the classical
theory of prefix codes. Although single- and two-channel prefix codes always
have decoding trees, the same cannot be said when there are more than two
channels. One question is of theoretical interest: Do there exist optimal
tree-decodable codes that are not optimal prefix codes? Existing literature,
which focused on generalizing single-channel results, covered little about
non-tree-decodable prefix codes since they have no single-channel counterparts.
In this work, we study the fundamental reason behind the non-tree-decodability
of prefix codes. By investigating the simplest non-tree-decodable structure, we
obtain a general sufficient condition on the channel alphabets for the
existence of optimal tree-decodable codes that are not optimal prefix codes.
