Prefix Tree Traversal

Haskell, 125 bytes

t=tail.p
p=g.break(=='[')
g(a,(_:t))=(:)&(map(a++).z)$t#[]
z[]=[""];z x=x
(']':u)#a=u:a
s#a=(#)&(a++)$p s
(g&f)(x:y)=g x$f y

The function is t (for traversal):

λ: t "cat[s[up[][]][]ch[e[r[]s[]]]a[maran[]comb[]pult[[]ing[]]]]"
["catsup","cats","cat","catcher","catches","catamaran","catacomb","catapult","catapulting"]
λ: t "[donut[][]cruller[]]"
["donut","","cruller"]
λ: t "[[[[[]]]]]"
[""]

Java, 206 bytes

Defines a function that accepts a string as an argument and returns a list of strings. For an added bonus it returns strings in the same order as the question does.

int c,i;List a(String a){String b=a.substring(c,c=a.indexOf(91,c));List d=new ArrayList();for(;a.charAt(++c)!=93;)d.addAll(a(a));if(d.isEmpty())d.add("");for(i=0;i<d.size();)d.set(i,b+d.get(i++));return d;}

Example usage:

class A{
    public static void main(String[] args){
        System.out.println(new A.a("cat[s[up[][]][]ch[e[r[]s[]]]a[maran[]comb[]pult[[]ing[]]]]"));
    }

    int c,i;List a(String a){String b=a.substring(c,c=a.indexOf(91,c));List d=new ArrayList();for(;a.charAt(++c)!=93;)d.addAll(a(a));if(d.isEmpty())d.add("");for(i=0;i<d.size();)d.set(i,b+d.get(i++));return d;}
}

Expanded:

int c, i;
List a(String a){
    String b = a.substring(c, c = a.indexOf(91, c));
    List d = new ArrayList();
    for(; a.charAt(++c) != 93 ;)
        d.addAll(a(a));
    if (d.isEmpty())
        d.add("");
    for (i = 0; i < d.size();)
        d.set(i, b + d.get(i++));
    return d;
}

I will add an explanation tomorrow.

Ruby, 119 115

t=['']
l=[0]
gets.chars{|c|c<?]?t<<''&&(l<<0)[-2]+=1:c<?^?(x=l.pop;t.pop==''&&(puts t*''if x<1;t[-1]='')):t[-1]<<c}

Example

Try it: http://ideone.com/NW0CNB

Description

The program gets the input from stdin and outputs the result to stdout.

It traverses the tree keeping the current branch in a stack. There's also a different stack, called weights which keeps track of the number of children of each node. This is needed in order to determine if a node is really a leaf, or it had children in the past.

The readable program:

stack = ['']
weights = [0]

gets.chars do |c|
  case c
  when '['
    weights[-1] += 1
    stack << ''
    weights << 0
  when ']'
    last_weight = weights.pop

    if stack.pop == ''
      puts stack.join if last_weight < 1
      stack[-1] = ''
    end
  else
    stack[-1] << c
  end
end