Coding k-ary trees for efficient loopless generation in lexicographic order

In this paper, e-words are used for coding k-ary trees with n internal nodes. The properties of e-words are discussed in depth, such as the necessary and sufficient condition of e-words, and based on the properties, a loopless algorithm is obtained to generate e-words for k-ary trees in lexicographic order, which is more efficient in both space and time than the previous algorithm. In addition, e-words can also be easily generated in lexicographic order by a recursive algorithm, and in the order with the Strong Minimal Change Property (SMCP) by a loopless algorithm.