3437：Tree Grafting

• there is one node designated as the root, denoted root(T);
  
• the remaining nodes are partitioned into subsets T1, T2, ..., Tm, each of which is also a tree (subtrees).

It is often more convenient to represent an ordered tree as a rooted binary tree, so that each node can be stored in the same amount of memory. The conversion is performed by the following steps:

1. remove all edges from each node to its children;
2. for each node, add an edge to its first child in T (if any) as the left child;
3. for each node, add an edge to its next sibling in T (if any) as the right child.

This is illustrated by the following:

         0                             0
       / | \                          /
      1  2  3       ===>             1
        / \                           \
       4   5                           2
                                      / \
                                     4   3
                                      \
                                       5

In most cases, the height of the tree (the number of edges in the longest root-to-leaf path) increases after the conversion. This is undesirable because the complexity of many algorithms on trees depends on its height.

You are asked to write a program that computes the height of the tree before and after the conversion.

The input is given by a number of lines giving the directions taken in a depth-first traversal of the trees. There is one line for each tree. For example, the tree above would give dudduduudu, meaning 0 down to 1, 1 up to 0, 0 down to 2, etc. The input is terminated by a line whose first character is #. You may assume that each tree has at least 2 and no more than 10000 nodes.

For each tree, print the heights of the tree before and after the conversion specified above. Use the format:

 Tree t: h1 => h2