Scott spectral gaps for trees are bounded

Abstract

Given a Borel class of trees, we show that there is a tree in that class whose Scott sentence is not too much more complicated than the definition of the class. In particular, if the class is definable by a $\Pi_\alpha$ sentence, then there is a model of Scott rank at most $\alpha + 2$. This gives another proof-and one that does not require first proving Vaught's conjecture for trees-of the fact that trees are not faithfully Borel complete.