Are duplicate keys allowed in the definition of binary search trees?

Question

I\'m trying to find the definition of a binary search tree and I keep finding different definitions everywhere.

Some say that for any given subtree the left child k

Answer 1

All three definitions are acceptable and correct. They define different variations of a BST.

Your college data structure's book failed to clarify that its definition was not the only possible.

Certainly, allowing duplicates adds complexity. If you use the definition "left <= root < right" and you have a tree like:

      3
    /   \
  2       4

then adding a "3" duplicate key to this tree will result in:

      3
    /   \
  2       4
    \
     3

Note that the duplicates are not in contiguous levels.

This is a big issue when allowing duplicates in a BST representation as the one above: duplicates may be separated by any number of levels, so checking for duplicate's existence is not that simple as just checking for immediate childs of a node.

An option to avoid this issue is to not represent duplicates structurally (as separate nodes) but instead use a counter that counts the number of occurrences of the key. The previous example would then have a tree like:

      3(1)
    /     \
  2(1)     4(1)

and after insertion of the duplicate "3" key it will become:

      3(2)
    /     \
  2(1)     4(1)

This simplifies lookup, removal and insertion operations, at the expense of some extra bytes and counter operations.