What does it mean for two binary trees to be isomorphic?

Question

What does it mean for two binary trees to be isomorphic? I've been looking online and I can't seem to find a clear explanation.

As far as I understand, two trees are isomorphic if they have the same shape. So I'm guessing two identical trees which can contain different values in the nodes.

Answer 1

I think your understanding is pretty much correct. If you ignore the values and just look at the structure, then for every node in the first tree there must be a corresponding node in the other tree and vice versa.

Naturally, both trees would have the same number of nodes. In addition, you could write a (super-naive) algorithm to confirm this isomorphism by attempting all possible mapping functions, and ensuring that for every node in the first tree that gets mapped to a node in the other, the corresponding mapping happens with the parent and with the two children. There are obviously efficient algorithms to check for this.

You may benefit from reading about graph isomorphism first; trees are a special (and easier to solve) case since they do not have cycles.

Answer 2

Isomorphic comes from the Greek "same shape" (like isobar is points with the same air pressure and polygon means "many sided") so your understanding is correct. But don't make the mistake of assuming shape in this case is a physical shape (like the tree has one root, one left node and one right node; see below for example). Mathematicians have their own language which only sometimes bears a relationship to English :-)

It's not just binary trees. In mathematics, two structures are isomorphic if their properties are preserved regardless of their expression (you can have a function that translates A to B and another from B to A without loss of information).

For your particular case, it's the information in the tree that's preserved. If that information is the sorted contents (and it should be if you're using a binary tree), then the tree doesn't have to be the same physical shape at all - the following two would be isomorphic:

  2      1
 / \      \
1   3      2
            \
             3

A sorted linked list (or sorted array, for that matter) is also isomorphic to those since, in that case, no information would be lost in the transformations between the two.

If the binary tree wasn't sorted, then the information would just be the contents and all the following would be isomorphic (that second last one's just a bag, the last is a list):

  2      1           2   3                   +---+  +---+  +---+
 / \      \         /     \      +-------+   | 3 |->| 1 |->| 2 |
1   3      2       1       2     | 1,3,2 |   +---+  +---+  +---+
            \     /         \    +-------+
             3   3           1

Of course, an unsorted tree is a bit of a waste, but that's not relevant to this particular discussion.