Time Complexity to find height of a BST
Your algorithm runs in linear time on all inputs. The algorithm visits each node of the tree exactly once, and does $O(1)$ work per node. Therefore it runs in time $\Theta(n)$, where $n$ is the number of nodes.

The argument above is better than using recurrences, since it is more immediate. It also shows that if you had an arbitrary tree (not necessarily binary), the algorithm would still run in linear time.