All the keys in a binary search tree should be greater than or equal to the root of the tree, is my understanding correct?

Question

A post claims 5 of the trees shown in the following figure are all binary search trees

I guess the author's making a mistake, since the 2 trees on the right ARE NOT binary search trees.

As an example, in the last tree pointed out by red rectangle, the key 1 is in 2's right subtree, which doesn't satisfy the binary-search-tree property.

Besides the figure above, the post also says

the BST is an ordered data structure that allows no duplicate values

which, I guess, is another mistake.

Correcting the bugs above, I'd like to make the following conclusion,

all the keys in a binary search tree should be greater than or equal to the root of the tree,

or more precisely, in a binary search tree, all the keys of a subtree should be greater than or equal to the root of the subtree,

is my understanding correct?

one of my references:

https://en.wikipedia.org/wiki/Binary_search_tree

Answer 1

You are right that both trees on the right are not binary trees but you are wrong that all keys of a subtree are greater than the key at the root. It is true only for the right subtree. The keys of the left subtree are all lesser than the keys of the root. That is the very definition of a BST.

Concerning duplicated values, depending on whether you want to use your BST for representing sets or multisets, you can allow them or not, that will not change much the complexity of the underlying operations. In the textbook Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein, duplicated values are allowed (see definition in Chapter 12).