P70 - Tree construction from a node string

We suppose that the nodes of a multiway tree contain single characters. In the depth-first order sequence of its nodes, a special character ^ has been inserted whenever, during the tree traversal, the move is a backtrack to the previous level.

By this rule, the tree in the figure opposite is represented as: afg^^c^bd^e^^^

Define the syntax of the string and write a function (tree string) to construct the tree when the string is given. Work with lists (instead of strings). Write also an inverse function.