How many binary trees possible with N nodes using Catalan Number concept

So I was understanding Catalan Number Concept and try to implement it in code using topDown DP approach, so the recurrence relation is from what I have learned,

f(N) signifies total count of binary search trees possible by N number of nodes

f(N) = f(i-1).f(N-i)

where, 
i = 1.....N (i signifies root node and traverse through all the nodes),
N is total number of nodes present in the binary tree,
f(i-1) signifies number of left subtrees possible when i is a root node,
f(N-i) signifies number of right subtrees possible when i is a root node.

there's a mathematical formulae by which we can find f(N) value,

2NcN/(N+1)

. And, total count of binary trees possible by N number of nodes,

(factorial of N) * f(N)

My Doubt:

what's the difference between binary trees and binary search trees? I feel like both are defines same meaning but there's a difference that I'm not aware of so help me.

Solution

What's the difference between binary trees and binary search trees?

There is no difference in the shape of such trees, but binary search trees put a constraint on the values that its nodes have, which doesn't exist for (just) binary trees:

The value of a node must not be less than any of the values in its left subtree
The value of a node must not be greater than any of the values in its right subtree

Another way to express this constraint is:

The in-order traversal of the values in the tree must be non-decreasing.

Practically this means that when you have been given:

A shape of a binary tree with 𝑛 nodes
A set of 𝑛 unique values that are present in the tree

...then there is only one possibly way to associate those values with the tree's nodes for it to be a binary search tree. If it does not have to be a binary search tree, then there are 𝑛! ways to make that association.

Conclusions:

Given 𝑛, then 𝐶_𝑛 (the 𝑛^th Catalan number) represents:

the number of binary trees with 𝑛 nodes without values (i.e. number of shapes of binary trees)
the number of binary search trees with 𝑛 distinct values

The number of binary trees with 𝑛 distinct values is 𝑛!𝐶_𝑛