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

I just want to add some more information to what @Robert Paulson answered.

Let's assume that node contains key & data. So nodes with the same key might contain different data.
(So the search must find all nodes with the same key)

1. left <= cur < right

2. left < cur <= right

3. left <= cur <= right

4. left < cur < right && cur contain sibling nodes with the same key.

5. left < cur < right, such that no duplicate keys exist.

1 & 2. works fine if the tree does not have any rotation-related functions to prevent skewness.
But this form doesn't work with AVL tree or Red-Black tree, because rotation will break the principal.
And even if search() finds the node with the key, it must traverse down to the leaf node for the nodes with duplicate key.
Making time complexity for search = theta(logN)

3. will work well with any form of BST with rotation-related functions.
But the search will take O(n), ruining the purpose of using BST.
Say we have the tree as below, with 3) principal.

         12
       /    \
     10     20
    /  \    /
   9   11  12 
      /      \
    10       12

If we do search(12) on this tree, even tho we found 12 at the root, we must keep search both left & right child to seek for the duplicate key.
This takes O(n) time as I've told.

4. is my personal favorite. Let's say sibling means the node with the same key.
We can change above tree into below.

         12 - 12 - 12
       /    \
10 - 10     20
    /  \
   9   11

Now any search will take O(logN) because we don't have to traverse children for the duplicate key.
And this principal also works well with AVL or RB tree.