Calculating the number of unique BST generatable from n keys, why is my number so large
You overcounted some of the trees and left out two other possibilities. In your Case 1, for example, there is only one possible BST of that form, namely the one with 2 in the root, 1 in the left subtree and 3 in the right subtree (the leftmost tree below). Remember that you have to maintain the binary search tree condition: a node with value $k$ must have each of the values in its left subtree less than or equal to $k$ and each of the values in the right subtree must be greater than or equal to $k$. As a consequence, a BST with node values $\{1, 2, \dotsc,n\}$ must have inorder traversal equal to $\langle1, 2, \dotsc,n\rangle$. When $n=3$, then, we'll have these five BSTS: