  A dichotomy result of Sevenster (2014) completely classified the quantifier
prefixes of regular Independence-Friendly (IF) logic according to the patterns
of quantifier dependence they contain. On one hand, prefixes that contain
"Henkin" or "signalling" patterns were shown to characterize fragments of IF
logic that capture NP-complete problems; all the remaining prefixes were shown
instead to be essentially first-order.
  In the present paper we develop the machinery which is needed in order to
extend the results of Sevenster to non-prenex, regular IF sentences. This
involves shifting attention from quantifier prefixes to a (rather general)
class of syntactical tree prefixes.
  We partially classify the fragments of regular IF logic that are thus
determined by syntactical trees; in particular, a) we identify three tree
prefixes that are neither signalling nor Henkin, and yet express NP-complete
problems and other second-order concepts; and b) we give more general criteria
for checking the first-orderness of an IF sentence.
