A 2-3 tree with no 3-nodes is a perfectly balanced binary search tree. So as just noted, for its height, h

      h <= log2(N)
   

where N is the number of Nodes. But if there are no 3-nodes, each node contains only 1 key.

So for such a perfectly balanced binary search tree

      h = O(log(N))
   

where h is the height and N is the number of keys