  Bruyere and Carton lifted the notion of finite automata reading infinite
words to finite automata reading words with shape an arbitrary linear order L.
Automata on finite words can be used to represent infinite structures, the
so-called word-automatic structures. Analogously, for a linear order L there is
the class of L-automatic structures. In this paper we prove the following
limitations on the class of L-automatic structures for a fixed L of finite
condensation rank 1+\alpha. Firstly, no scattered linear order with finite
condensation rank above \omega^(\alpha+1) is L-\alpha-automatic. In particular,
every L-automatic ordinal is below \omega^\omega^(\alpha+1). Secondly, we
provide bounds on the (ordinal) height of well-founded order trees that are
L-automatic. If \alpha is finite or L is an ordinal, the height of such a tree
is bounded by \omega^{\alpha+1}. Finally, we separate the class of
tree-automatic structures from that of L-automatic structures for any ordinal
L: the countable atomless boolean algebra is known to be tree-automatic, but we
show that it is not L-automatic.
