Problem:
Problem F: Dr. Podboq or: How We Became Asymmetric

After long studying how embryos of organisms become asymmetric during
their development, Dr. Podboq, a famous biologist, has reached his new
hypothesis. Dr. Podboq is now preparing a poster for the coming
academic conference, which shows a tree representing the development
process of an embryo through repeated cell divisions starting from one
cell. Your job is to write a program that transforms given trees into
forms satisfying some conditions so that it is easier for the audience
to get the idea.

A tree representing the process of cell divisions has a form described
below.

The starting cell is represented by a circle placed at the top.

Each cell either terminates the division activity or divides into
two cells. Therefore, from each circle representing a cell, there are
either no branch downward, or two branches down to its
two child cells.

Below is an example of such a tree.

Figure F-1: A tree representing a process of cell divisions

According to Dr. Podboq's hypothesis, we can determine which cells
have stronger or weaker asymmetricity by looking at the structure of
this tree representation. First, his hypothesis defines "left-right
similarity" of cells as follows:

The left-right similarity of a cell that did not divide further is
0.

For a cell that did divide further, we collect the partial
trees starting from its child or descendant cells, and count how
many kinds of structures they have. Then, the left-right similarity of
the cell is defined to be the ratio of the number of structures
that appear both in
the right child side and the left child side. We regard two trees
have the same structure if we can make them have exactly the same shape by
interchanging two child cells of arbitrary cells.

For example, suppose we have a tree shown below:

Figure F-2: An example tree

The left-right similarity of the cell A is computed as follows.
First, within the descendants of the cell B, which is the left child
cell of A, the following three kinds of structures appear. Notice that
the rightmost structure appears three times, but when we count the number
of structures, we count it only once.

Figure F-3: Structures appearing within the descendants of the cell B

On the other hand, within the descendants of the cell C, which is the
right child cell of A, the following four kinds of structures appear.

Figure F-4: Structures appearing within the descendants of the cell C

Among them, the first, second, and third ones within the B side are
regarded as the same structure as the second, third, and fourth ones
within the C side, respectively. Therefore, there are four structures
in total, and three among them are common to the left side and the
right side, which means the left-right similarity of A is 3/4.

Given the left-right similarity of each cell, Dr. Podboq's hypothesis
says we can determine which of the cells X and Y has
stronger asymmetricity by the following rules.

If X and Y have different left-right similarities, the
one with lower left-right similarity has stronger asymmetricity.

Otherwise, if neither X nor Y has child cells, they
have completely equal asymmetricity.

Otherwise, both X and Y must have two child cells. In this case,
we compare the child cell of X with stronger (or equal)
asymmetricity (than the other child cell of X) and the child
cell of Y with stronger (or equal) asymmetricity (than the other child cell
of Y), and the one having a child with stronger asymmetricity
has stronger asymmetricity.

If we still have a tie, we compare the other child cells of X
and Y with weaker (or equal) asymmetricity, and the one having a child with
stronger asymmetricity has stronger asymmetricity.

If we still have a tie again, X and Y have
completely equal asymmetricity.

When we compare child cells in some rules above, we recursively apply
this rule set.

Now, your job is to write a program that transforms a given tree
representing a process of cell divisions, by interchanging two child cells
of arbitrary cells, into a tree where the following conditions are
satisfied.

For every cell X which is the starting cell of the given
tree or a left child cell of some parent cell, if X has two
child cells, the one at left has stronger (or equal) asymmetricity than the one
at right.

For every cell X which is a right child cell of some parent
cell, if X has two child cells, the one at right has stronger (or equal)
asymmetricity than the one at left.

In case two child cells have equal asymmetricity, their order is
arbitrary because either order would results in trees of the same shape.

For example, suppose we are given the tree in Figure F-2. First we
compare B and C, and because B has lower left-right similarity, which means stronger asymmetricity, we
keep B at left and C at right. Next, because B is the left child cell
of A, we compare two child cells of B, and the one with stronger
asymmetricity is positioned at left. On the other hand, because C is
the right child cell of A, we compare two child cells of C, and the one
with stronger asymmetricity is positioned at right. We examine the
other cells in the same way, and the tree is finally transformed into
the tree shown below.

Figure F-5: The example tree after the transformation

Please be warned that the only operation allowed in the transformation
of a tree is to interchange two child cells of some parent cell. For
example, you are not allowed to transform the tree in Figure F-2 into the tree
below.

Figure F-6: An example of disallowed transformation

Input

The input consists of n lines (1≤n≤100) describing
n trees followed by a line only containing a single zero which
represents the end of the input. Each tree includes at least 1 and at
most 127 cells. Below is an example of a tree description.

((x (x x)) x)

This description represents the tree shown in Figure F-1. More
formally, the description of a tree is in either of the following two formats.

"(" <description of a tree starting at the left child> <single space> <description of a tree starting at the right child> ")"

or

"x"

The former is the description of a tree whose starting cell has two
child cells, and the latter is the description of a tree whose starting
cell has no child cell.

Output

For each tree given in the input, print a line describing the result
of the tree transformation. In the output, trees should be described in
the same formats as the input, and the tree descriptions must
appear in the same order as the input. Each line should have no extra
character other than one tree description.

Sample Input

(((x x) x) ((x x) (x (x x))))
(((x x) (x x)) ((x x) ((x x) (x x))))
(((x x) ((x x) x)) (((x (x x)) x) (x x)))
(((x x) x) ((x x) (((((x x) x) x) x) x)))
(((x x) x) ((x (x x)) (x (x x))))
((((x (x x)) x) (x ((x x) x))) ((x (x x)) (x x)))
((((x x) x) ((x x) (x (x x)))) (((x x) (x x)) ((x x) ((x x) (x x)))))
0

Output for the Sample Input

((x (x x)) ((x x) ((x x) x)))
(((x x) ((x x) (x x))) ((x x) (x x)))
(((x ((x x) x)) (x x)) ((x x) ((x x) x)))
(((x ((x ((x x) x)) x)) (x x)) ((x x) x))
((x (x x)) ((x (x x)) ((x x) x)))
(((x (x x)) (x x)) ((x ((x x) x)) ((x (x x)) x)))
(((x (x x)) ((x x) ((x x) x))) (((x x) (x x)) (((x x) (x x)) (x x))))

Solution in Ruby (judge verdict: Accepted)

require 'set'

### global variables

$node = [Rational(0), Set.new(['x']), 'x']

### subroutines

def asym?(tr0, tr1)
  sim0, set0, sk0, tr00, tr01 = tr0
  sim1, set1, sk1, tr10, tr11 = tr1

  # 1
  return sim1 <=> sim0 if sim0 != sim1

  # 2
  return 0 if tr00.nil? && tr01.nil? && tr10.nil? && tr11.nil?

  if asym?(tr00, tr01) >= 0
    tr0s = tr00
    tr0w = tr01
  else
    tr0s = tr01
    tr0w = tr00
  end

  if asym?(tr10, tr11) >= 0
    tr1s = tr10
    tr1w = tr11
  else
    tr1s = tr11
    tr1w = tr10
  end

  # 3
  cmp_s = asym?(tr0s, tr1s)
  return cmp_s if cmp_s != 0

  # 4
  cmp_w = asym?(tr0w, tr1w)
  return cmp_w if cmp_w != 0

  # 5
  0
end

# <tree> := 'x' | '(' <tree> ' ' <tree> ')'

def parse_tree(str)
  if str[$pos] == 'x'
    $pos += 1
    return $node
  end

  $pos += 1
  tr0 = parse_tree(str)
  $pos += 1
  tr1 = parse_tree(str)
  $pos += 1

  set0 = tr0[1]
  set1 = tr1[1]

  set_c = set0 & set1
  set_u = set0 | set1
  sim = Rational(set_c.length, set_u.length)

  cmp = asym?(tr0, tr1)
  if cmp < 0
    tmp = tr0; tr0 = tr1; tr1 = tmp
  end

  sk = [tr0[2], tr1[2]]
  [sim, set_u | [sk], sk, tr0, tr1]
end

def arrange_tree(tr, rev = false)
  return 'x' if tr == $node

  trl = tr[3]
  trr = tr[4]

  if rev
    tmp = trl; trl = trr; trr = tmp
  end

  atrl = arrange_tree(trl, false)
  atrr = arrange_tree(trr, true)

  "(#{atrl} #{atrr})"
end

### main

loop do
  str = gets.strip
  break if str == '0'

  $pos = 0
  tr = parse_tree(str)
  #p tr[2]
  puts arrange_tree(tr)
end