C# and Db are one and the same, as shown in the first measure of the example. If you trill the two, as seen in the second measure, you'd expect the two trills to be the same. However, a trill is played by rapidly alternating between the root note and the note directly above it in that key. For C#, it would be D (assuming C Major), and for Db, it would be E. I've expanded the trills as it seems they should be played in the third and fourth measures.
This doesn't make sense to me as the two root notes are the same. Do I have it right, or is there some rule that I'm not aware of concerning trilled notes?