A family of (k, m)-ary trees was firstly introduced by Du and Liu when they studied hook length polynomial for plane trees. Recently, Amani and Nowzari-Dalini presented a generation algorithm to produce (k, m)-ary trees of order n encoding by Z-sequences in reverse lexicographic order. In this paper, we propose a loopless algorithm to generate all such Z-sequences in Gray code order. Hence, the worst-case time complexity of generating one Z-sequence is \({\mathcal {O}}(1)\), and the space requirement of our algorithm is \(2n+{\mathcal {O}}(1)\). Moreover, based on this ordering, we also provide ranking and unranking algorithms. The ranking algorithm can be run in \({\mathcal {O}}(\max \{kmn,n^2\})\) time and \({\mathcal {O}}(kmn)\) space, whereas the unranking algorithm requires \({\mathcal {O}}(kmn^2)\) time and space.

Du和Liu在研究平面树的钩长多项式时首先引入了（k，  m）元树族。最近，Amani和Nowzari-Dalini提出了一种生成算法，以按字典顺序的Z序列生成n阶的（k，  m）进制树。在本文中，我们提出了一种无环算法，以格雷码顺序生成所有此类Z序列。因此，生成一个Z序列的最坏情况下的时间复杂度为\（{\ mathcal {O}}（1）\），而我们算法的空间要求为\（2n + {\ mathcal {O}}（1 ）\）。此外，基于此排序，我们还提供了排名和排名算法。排序算法可以在\（{\ mathcal {O}}（\ max \ {kmn，n ^ 2 \}））\）时间和\（{\ mathcal {O}}（kmn）\）空间中运行，而解等级算法需要\（{\ mathcal {O}}（kmn ^ 2）\）的时间和空间。