We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal.

翻译：我们研究二元字母表上的前缀自由码与一类无标号有根树（称为对称树）之间的结构关系。我们建立了前缀自由码与对称树之间的典范对应，该对应不仅保持码字的长度，还保留了额外的交换结构。利用这一对应，我们给出了与交换等价猜想相关的结果。我们证明：对每个码，存在一个前缀自由码，使得对每个固定的字长，由特定符号出现所决定的2的幂次之和相等。