如何在Java中生成树？

Question

So how do I tackle this problem? I need to a program that reads a positive integer n from standard input and writes to standard output a representation of all distinct rooted, ordered, labeled trees on the set {1,2,3....n} of vertices.

For the output, I need to use the following linear textual representation L(t) of a tree t :

 If t is empty then L(t) = ().
 If t has root n and children, in sibling order, C = (c1; c2; : : : ; ck), then
 L(t) = (n, (L(c1), L(c2), : : :, L(ck)))
 where, L(ci) denotes the linear textual representation of the subtree rooted at
  child ci. There is a single space after each comma in the representation.

The output should contain the representation of one tree on each line and should be sorted by lexicographic order on the linear representations viewed as strings. The output should contain nothing else (such as spurious newline characters, prompts, or informative messages). Sample inputs and outputs for n = 1; 2; appear below.

enter code here

Input: 1
Output:
(1, ())
Input: 2
Output:
(1, ((2, ()))) 
(2, ((1, ())))

enter code here

Any help will be largely appreciated. I just need to be steered to a direction. Right now, I'm completely stumped :(

Answer 1

You can generate trees recursively. Start with a root. The root can have 0, 1, 2 ... (m - 1) children, where m is the number of vertices you have left to place. Start by placing (m - 1) vertices under the root, and then go down all the way to 0. You'll "place" these vertices recursively, so placing a vertex as child under the root means calling the same method again, but the maximum number of children will be a bit less this time.

You'll get two stopping criteria for the recursion:

• You've placed all N vertices. You need to output the current tree then with your yet-to-define L(t) function, then backtrack to try different trees.
• The algorithm gave all leaf vertices a degree of 0 and you haven't placed n vertices yet. Don't output this tree since it is invalid, but backtrack.

The algorithm is finished after it tries to give the root node 0 children.

As for the output L(t) function, it seems to suffice to do a depth-first tree traversal. Recursive is easiest to program (as this seems to be a practical assignment of some kind, that's probably what they want you to do). If you need speed, look for a non-recursive depth-first tree traversal algorithm on Wikipedia.