Is there a name for this traversal-optimized representation of a binary tree data structure?
This representation is known as a "zipper" and is described by Gérard Huet in his paper "The Zipper":


  The basic idea is simple: the tree is turned inside-out like a returned glove,
  pointers from the root to the current position being reversed in a path structure. The current location holds both the downward current subtree and the upward path. All navigation and modification primitives operate on the location structure. Going up and down in the structure is analogous to closing and opening a zipper in a piece of clothing, whence the name.