How to write recurrence relation for the following scenario?
Hints:

Can you make a guess?  If you can make a guess, can you prove it?

What do you know about the height of a balanced search tree?

If the entire tree has $n$ nodes, what can you say about the number of nodes in the left subtree?  What is the smallest it could be?  The largest it could be?  What about the right subtree? 

Can you derive a recurrence relation?

Imagine you treat the cost of computing $g(x)$ to be measured in dollars, and you charge \$1 to each node in the right/left subtree (whichever is smaller) to account for the cost of computing $g(x)$.  For any given node, what can you say about how many dollars it is charged?

If you know about amortized analysis, you might review that material.

It's not clear to me what $k$ represents in your proposed approach or where you got that recurrence relation.  Why don't you try harder to work out what recurrence relation might be suitable and what its solution is?