Iterating over a Binary Tree with O(1) Auxiliary Space

Question

Is it possible to iterate over a binary tree in O(1) auxiliary space (w/o using a stack, queue, etc.), or has this been proven impossible? If it is possible, how can it be

Answer 1

Pointers from nodes to their ancestors can be had with no (well, two bits per node) additional storage using a structure called a threaded tree. In a threaded tree, null links are represented by a bit of state rather than a null pointer. Then, you can replace the null links with pointers to other nodes: left links point to the successor node in an inorder traversal, and right links to the predecessor. Here is a Unicode-heavy diagram (X represents a header node used to control the tree):

                                         ╭─┬────────────────────────────────────────╮
   ╭─────────────────────────▶┏━━━┯━━━┯━━▼┓│                                        │
   │                        ╭─╂─  │ X │  ─╂╯                                        │ 
   │                        ▼ ┗━━━┷━━━┷━━━┛                                         │
   │                    ┏━━━┯━━━┯━━━┓                                               │
   │               ╭────╂─  │ A │  ─╂──╮                                            │
   │               ▼    ┗━━━┷━━━┷━━━┛  │                                            │    
   │        ┏━━━┯━━━┯━━━┓    ▲         │        ┏━━━┯━━━┯━━━┓                       │
   │      ╭─╂─  │ B │  ─╂────┤         ├────────╂─  │ C │  ─╂───────╮               │
   │      ▼ ┗━━━┷━━━┷━━━┛    │         ▼        ┗━━━┷━━━┷━━━┛       ▼               │  
   │┏━━━┯━━━┯━━━┓ ▲          │   ┏━━━┯━━━┯━━━┓       ▲         ┏━━━┯━━━┯━━━┓        │
   ╰╂─  │ D │  ─╂─╯          ╰───╂   │ E │  ─╂╮      │        ╭╂─  │ F │  ─╂╮       │ 
    ┗━━━┷━━━┷━━━┛                ┗━━━┷━━━┷━━━┛▼      │        ▼┗━━━┷━━━┷━━━┛▼       │
                                    ▲ ┏━━━┯━━━┯━━━┓  │ ┏━━━┯━━━┯━━━┓ ▲ ┏━━━┯━━━┯━━━┓│
                                    ╰─╂─  │ G │   ╂──┴─╂─  │ H │  ─╂─┴─╂─  │ J │  ─╂╯
                                      ┗━━━┷━━━┷━━━┛    ┗━━━┷━━━┷━━━┛   ┗━━━┷━━━┷━━━┛

Once you have the structure, doing an inorder traversal is very, very easy:

Inorder-Successor(p)
    p points to a node.  This routine finds the successor of p in
    an inorder traversal and returns a pointer to that node

    q ← p.right
    If p.rtag = 0 Then
        While q.ltag = 0 Do
            q ← q.left
        End While
    End If

    Return q
    

Lots more information on threaded trees can be found in Art of Computer Programming Ch.2 §3.1 or scattered around the Internet.