Question

I know that from a general tree you can construct a unique binary tree, but is the reverse true? i.e can you get a unique general tree from a binary tree?

Answer 1

Yes. The following transformation is reversible:

Given a general tree with ordered but not indexed children,
encode the first child as the left child of its parent, and each other node as a right child of its (former) sibling.

The reverse is:
Given a binary tree with distinguished left and right children, read the left child of a node as its first child and the right child as its next sibling.

So, the following tree

  a
 /|\
b c d

is encoded as

  a
 /
b
 \
  c
   \
    d

while the following tree

   a
  / \
 b   c
 |
 d

is encoded as

     a
    /
   b
  / \
 d   c

(read: d is the first child of b, c is the sibling of a).

Note that you can encode any rooted forest (with ordered components, otherwise the representation is not unique) by assigning a sibling to the root, so this

  a
 / \
b   c
 \   \
  d   e

would be read as

  a   c e
 / \
b   d

---

here is another method to get a unique general (undirected) tree from a binary tree:

• a vertex binary tree may have 0…3 graph neighbors.
• append 12 nodes to the root
• append 8 nodes to each left child
• append 4 nodes to each right child

this operation is reversible:

• label the node with at least 12 neighbors “root”. If not unique, fail.
• label each node with 8..11 neighbors “left”.
• label each node with 4..7 neighbors “right”.
• remove all leaves
• orient all edges away from the root
• if any node has more than one left child or more than one right child, fail.

So,

• There is a bijection between ordered rooted trees and binary trees (first and second algorithm).
• Since any general tree can be arbitrarily rooted, there is a injection from general (directed or undirected) trees to binary trees.
• There is an injection from binary trees to general undirected trees (third algorithm)
• Since there is an injection from binary trees to general trees and back, there must exist a bijection between general (directed or undirected) trees and binary trees.