  A premonoidal category is equipped only with a bifunctor and a natural
isomorphism for associativity. We introduce a (deformation) natural
automorphism representing the deviation from the Pentagon condition. We uncover
a binary tree representation for all diagrams involving associativity natural
isomorphisms and (deformation) natural automorphisms and provide a link to
permutations and linear orderings. This leads to other notions of
premonoidalness. We define these notions and prove coherence results for each.
