Forests and the W Construction

A modified definition of the category of forests by Costello and expounded upon by Getzler is given. This language is used to prove some cofibration properties of the W construction for colored operads, following the work of Berger-Moerdijk. An application is given to the swiss cheese operad. The language of forests is convenient for showing that the swiss cheese operad is equivalent to its free degree 0 and 1 pieces.