Question

This is homework question, so I just need help may be yes/No and few comment will be appreciated!

• Prove: Arbitrary tree (NON binary tree) can be converted to equivalent binary decision tree.

My answer:
Every decision can be generated just using binary decisions. Hence that decision tree too.
I don’t know formal proof. Its like I can argue with Entropy(Gain actually) for that node will be E(S) – E(L) – E(R). And before that may be it is E(S) – E(Y|X=t1) – E(Y|X=t2) – and so on.

But don’t know how to say?!

Answer 1

You can give a constructive proof of something like this, demonstrating how to convert an arbitrary decision tree into a binary decision tree.

Imagine that you are sitting at node A, and you have a choice of traversing to B, C, and D based on whether or not your example satisfies requirements B, C or D. If this is a proper decision tree, B, C and D are mutually exclusive and cover all cases.

A -> B
  -> C
  -> D

Since they’re mutually exclusive, you could imagine splitting your tree into a binary decision: B or not B; on the not B branch, we know that either C or D has to be true, since B, C, and D were mutually exclusive and cover all cases. In other words:

A -> B
  -> ~B
  ---> C
  ---> D

Then you can copy whatever was going to go after B on to the branch that follows B, performing the same simplification. Same for C and D.