Question

I don’ get it? Shouldn’t they be different as ordered trees too? because the ordering is different

Answer 1

In an ordered tree, child nodes are ordered from left to right. They are not ordered with respect to their parent node (or, alternatively, you could think of the parent node always coming first). If there is only one child, there is only one child.

In a binary tree, there is an (optional) left child and an (optional) right child. If there is only one child, it could be either a left child or a right child, and the two cases are different. Alternatively, you can think of the parent node coming between the child nodes, so you can distinguish a child node which comes before the parent from a child node which comes after the parent.

There is a homomorphism between ordered trees and binary trees with the same number of nodes: that is, every ordered tree corresponds uniquely to a binary tree. To find the binary tree corresponding to a ordered tree: make the left child of each node in the binary tree point to the leftmost child of the node in the ordered tree, and make the right child of each node in the binary tree point to the sibling to the right of the node in the ordered tree. (It should be obvious how to reverse the process, so that you can see that every binary tree corresponds uniquely to an ordered tree.)

Consequently, the number of binary trees with k nodes is the same as the number of ordered tree with k nodes.