Are there any theorems/formulas that apply to the height of comparison trees?
Basically, we just need to look at the worst case comparison number, a worst case comparison number is when we  compare 1 with 2...N, then 2 with 3...N, etc. that will take (N^2 - N)/2 steps (arithmetic progression ((N-1)*(N-1+1))/2). So that's the height of that tree.

So for N = 3 we get (3^3 - 3)/2 = (9 - 3)/2 = 3. For N = 4 we have (4^2 - 4)/2 = 6.