Help me understand Inorder Traversal without using recursion

Question

I am able to understand preorder traversal without using recursion, but I'm having a hard time with inorder traversal. I just don't seem to get it, perhaps, because I haven't understood the inner working of recursion.

This is what I've tried so far:

def traverseInorder(node):
    lifo = Lifo()
    lifo.push(node)
    while True:
        if node is None:
            break
        if node.left is not None:
            lifo.push(node.left)
            node = node.left
            continue
        prev = node
        while True:
            if node is None:
                break
            print node.value
            prev = node
            node = lifo.pop()
        node = prev
        if node.right is not None:
            lifo.push(node.right)
            node = node.right
        else:
            break

The inner while-loop just doesn't feel right. Also, some of the elements are getting printed twice; may be I can solve this by checking if that node has been printed before, but that requires another variable, which, again, doesn't feel right. Where am I going wrong?

I haven't tried postorder traversal, but I guess it's similar and I will face the same conceptual blockage there, too.

Thanks for your time!

P.S.: Definitions of Lifo and Node:

class Node:
    def __init__(self, value, left=None, right=None):
        self.value = value
        self.left = left
        self.right = right

class Lifo:
    def __init__(self):
        self.lifo = ()
    def push(self, data):
        self.lifo = (data, self.lifo)
    def pop(self):
        if len(self.lifo) == 0:
            return None
        ret, self.lifo = self.lifo
        return ret

Answer 1

Start with the recursive algorithm (pseudocode) :

traverse(node):
  if node != None do:
    traverse(node.left)
    print node.value
    traverse(node.right)
  endif

This is a clear case of tail recursion, so you can easily turn it into a while-loop.

traverse(node):
  while node != None do:
    traverse(node.left)
    print node.value
    node = node.right
  endwhile

You're left with a recursive call. What the recursive call does is push a new context on the stack, run the code from the beginning, then retrieve the context and keep doing what it was doing. So, you create a stack for storage, and a loop that determines, on every iteration, whether we're in a "first run" situation (non-null node) or a "returning" situation (null node, non-empty stack) and runs the appropriate code:

traverse(node):
  stack = []
  while !empty(stack) || node != None do:
    if node != None do: // this is a normal call, recurse
      push(stack,node)
      node = node.left
    else // we're returning: pop and print the current node
      node = pop(stack)
      print node.value
      node = node.right
    endif
  endwhile

The hard thing to grasp is the "return" part: you have to determine, in your loop, whether the code you're running is in the "entering the function" situation or in the "returning from a call" situation, and you will have an if/else chain with as many cases as you have non-terminal recursions in your code.

In this specific situation, we're using the node to keep information about the situation. Another way would be to store that in the stack itself (just like a computer does for recursion). With that technique, the code is less optimal, but easier to follow

traverse(node):
  // entry:
  if node == NULL do return
  traverse(node.left)
  // after-left-traversal:
  print node.value
  traverse(node.right)

traverse(node):
   stack = [node,'entry']
   while !empty(stack) do:
     [node,state] = pop(stack)
     switch state: 
       case 'entry': 
         if node == None do: break; // return
         push(stack,[node,'after-left-traversal']) // store return address
         push(stack,[node.left,'entry']) // recursive call
         break;
       case 'after-left-traversal': 
         print node.value;
         // tail call : no return address
         push(stack,[node.right,'entry']) // recursive call
      end
    endwhile 

Answer 2

I think part of the problem is the use of the "prev" variable. You shouldn't have to store the previous node you should be able to maintain the state on the stack (Lifo) itself.

From Wikipedia, the algorithm you are aiming for is:

1. Visit the root.
2. Traverse the left subtree
3. Traverse the right subtree

In pseudo code (disclaimer, I don't know Python so apologies for the Python/C++ style code below!) your algorithm would be something like:

lifo = Lifo();
lifo.push(rootNode);

while(!lifo.empty())
{
    node = lifo.pop();
    if(node is not None)
    {
        print node.value;
        if(node.right is not None)
        {
            lifo.push(node.right);
        }
        if(node.left is not None)
        {
            lifo.push(node.left);
        }
    }
}

For postorder traversal you simply swap the order you push the left and right subtrees onto the stack.

Answer 3

State can be remembered implicitly,

traverse(node) {
   if(!node) return;
   push(stack, node);
   while (!empty(stack)) {
     /*Remember the left nodes in stack*/
     while (node->left) {
        push(stack, node->left);
        node = node->left;
      }

      /*Process the node*/
      printf("%d", node->data);

      /*Do the tail recursion*/
      if(node->right) {
         node = node->right
      } else {
         node = pop(stack); /*New Node will be from previous*/
      }
    }
 }

Answer 4

class Node { public int value; public Node small; public Node large; public Node parent; public bool visited; } public void PostIterative(Node _root) { System.Collections.Stack stk = new System.Collections.Stack();

        stk.Push(_root);

        while (stk.Count != 0)
        {
            Node n = (Node)stk.Peek();
            if (n.small != null && n.small.visited == false) stk.Push(n.small);
            else if (n.large != null && n.large.visited == false) stk.Push(n.large);
            else
            {
                Console.WriteLine(n.value);
                n.visited = true;
                stk.Pop();
            }
        }
    }