I've ridden the katy several times (I live in St. Louis).

I did a 100-miler on the Katy last year with slick road tires (700x35). I had a flat but it didn't appear to be caused by trail conditions. I did another 100-miler this year with semi-smooth tires (Continental Top Toures - 700x32) and had no problems at all.

I think the biggest factor with tires on the katy trail is width. The gravel gets a little soft in places, especially near the shoulder, and I think a wide tire would be more stable than a narrow. By wide I mean wider than a standard 700x23 road tire.

Also there are long stretches where they've allowed farm equipment to use the trail, and the surface is really rough. Again, a wide tire floats over that better than narrow.

So, in my view, slick is OK, and wide is best.

NOTE: If the weather has been dry be prepared for dusty conditions. It can be a real bear breathing in that dust on a hot, dry day.