Are empty Binary Search Trees valid?

Question

I have two questions regarding binary search trees, both about empty trees.

1. Is an empty tree (null) valid?
2. Is a root node without children valid?

Answer 1

I think you're mixing together apples and oranges:

1. null is a value in some programming languages, and it is related upon the concrete representation of your implementation of the data structure
2. "Empty" is a property of the abstract data structure that we call "binary search tree"

Now, a tree is an ordered set: nothing more, nothing less. A set can be empty, of course! This means that, in your implementation, something similar to:

MyTree tree = null

represent an empty tree? Well, it depends upon your model. You can think, for ex., that an empty subtree should be represented by a node without value and with the references to the leafs nullified: in that model, a null pointer is meaningless from a logical perspective. But this is only one approach! The sentinel-based approach is a joy to program with, but it's memory expansive: you can then model empty nodes with just null. In that case, a null pointer can be an empty tree.