Tree-automatic scattered linear orders

We study tree-automatic linear orders on regular tree languages. We first show that there is no tree-automatic scattered linear order, and in particular no well-order, on the set of all finite labeled trees. This also follows from results of Gurevich-Shelah [8] and Carayol-Löding [4]. We then show that a regular tree language admits a tree-automatic scattered linear order if and only if all trees are included in a subtree of the full binary tree with finite tree-rank. As a consequence of this characterization, we obtain an algorithm which, given a regular tree language, decides if the tree language can be well-ordered by a tree automaton. Finally, we connect tree automata with automata on ordinals and determine sharp lower and upper bounds for tree-automatic well-orders on natural examples of regular tree languages. Our proofs use elementary techniques of automata theory.