The cutthroat spawn in the tributary streams and migrate to the Lake. This "Adfluvial" lifecycle allows them to grow bigger then if they were just residents of the streams.

Lake trout eat the cutthroats (all sizes), and also eat food that would feed the cuts.

In other lakes there are always compromises made, but generally there is often a large enough prey base to feed Lakers and other fish, or at least where Lakers have been introduced, nobody noticed a difference (few users, not enough info, met the objectives to start a fishery, etc). Lakers weren't introduced to Yellowstone till the mid 80's, and people already have fishery expectations... cutthroats.
I'm not sure what the natural prey base was like in the Lake, but I'm sure the balance has been destroyed and Lakers are preying more on their own young than anything else (my speculation only).

In 2010 there will be the next big trout conference in West Yellowstone, if anybody is interested, it will be a great conference (science).
http://www.wildtroutsymposium.com/
I'm sure the Cutthroat/Lake trout interactions will be one of the topics.