reconstructing a tree from its preorder and postorder lists

Question

Consider the situation where you have two lists of nodes of which all you know is that one is a representation of a preorder traversal of some tree and the other a represent

Answer 1

You cannot use only one list, because you'll get no sense of the depth of the tree. Thus, you definitely require two or more lists.

Here's my attempt at a solution:

Use your preorder traversal as a means of knowing the ordering of the data. This makes sense because you know the first node is the top, and you know that data further to the left of the traversal belongs to the left of the tree, etc.

Your post order traversal can determine the depth of the tree. For example, let's say I have a structure like this:

      1
  2   5   6
 3 4  7

Where 2 is the parent of 3 and 4, and 5 is the parent of 7.

Preorder: 1 2 3 4 5 7 6
Postorder: 3 4 2 7 5 6 1

We know we start with 1, because it is the first node in the preorder traversal. Then we look at the next number, 2. In the post order, because the number 2 comes BEFORE node 1, we know that 2 has to be a child of 1. Next we look at 3. 3 comes before 2, and thus 3 is a child of 2. 4 is before 2 but after 3, so we know 4 is a child of 2 but NOT a child of 3. Etc.

Now, this may not work if the nodes are not unique, but at the very least its a start to a solution.

Edit: The order of the children is preserved with this solution, simply due to knowing the ordering of the nodes via the preorder traversal, and then knowing the structure via the postorder traversal.

Edit2: The proof may lie here: http://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel2%2F215%2F626%2F00017225.pdf%3Farnumber%3D17225&authDecision=-203

I think you need to purchase the document, however...

Here is a written proof presented to be a solution:

http://www14.informatik.tu-muenchen.de/lehre/2007WS/fa-cse/tutorials/tutorial09-solutions.pdf