Specializing trees and answer to a question of Williams

We show that if cf(2-0) =-1, then any nontrivial-1-closed forcing notion of size ≤ 2-0 is forcing equivalent to Add(-1, 1), the Cohen forcing for adding a new Cohen subset of ω1. We also produce, relative to the existence of suitable large cardinals, a model of ZFC in which 2-0 =-2 and all-1-closed forcing notion of size ≤ 2-0 collapse-2, and hence are forcing equivalent to Add(-1, 1). These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman-Magidor-Shelah by showing that it is consistent that every partial order which adds a new subset of-2, collapses-2 or-3.