Forest-regular languages and tree-regular languages

De nition 2.1 (forest). A forest over is: (1) (the null forest), (2) ahui, where a is a symbol in and u is a forest, or (3) uv, where u and v are forests. The set of forests over is denoted by F . For any forest u; v; w 2 F ; u(vw) = (uv)w and u = u = u. We abbreviate ah i as a. Remark. Since abc = ah ibh ich i : : : , a string is also a forest. De nition 2.2 (tree). A tree is a forest of the form ahui. The set of trees over is denoted by T . De nition 2.3 (forest width). The width of a forest u, denoted juj, is the number of trees at the top level of u. That is, j j = 0; jahuij = 1, and juvj = juj+ jvj. De nition 2.4 (forest domain). We assign to each u 2 F a subset of f1; 2; 3; : : :g , denoted Dom(u), such that: (1) if u = , then Dom(u) = ;, (2) if u = ahvi, then Dom(u) = f1g [ f1 v1v2 : : : vk j k 0; v1v2 : : : vk 2 Dom(v)g,