Tree-automatic scattered linear orders

Tree-automatic linear orders on regular tree languages are studied. It is shown that there is no tree-automatic scattered linear order, and therefore no tree-automatic well-order, on the set of all finite labeled trees, and that a regular tree language admits a tree-automatic scattered linear order if and only if for some n, no binary tree of height n can be embedded into the union of the domains of its trees. Hence the problem whether a given regular tree language can be ordered by a scattered linear order or a well-order is decidable. Moreover, sharp bounds for tree-automatic well-orders on some regular tree languages are computed by connecting tree automata with automata on ordinals. The proofs use elementary techniques of automata theory. (C) 2016 Elsevier B.V. All rights reserved.