If you swing over to our "Genre" section, you'll see we peg Cat's Cradle as a tragicomedy. Although it has both elements of tragedy and comedy, its plot structure is definitely that of a tragedy. This is mostly due to the fact that the story ends in the apocalypse, and, hey, there's nothing more tragic than the end of all life on Earth, right?
But now we're getting ahead of ourselves. Let's plot this thing:
The tragedy begins with what's called the Anticipation Stage. Here, the protagonist turns his focus toward some course of action. In Cat's Cradle, John turns to writing a book about the atomic bomb. His focus is on collecting information for that book, but while collecting said info, he learns about the history of the Hoenikker clan as well as the deadly ice-nine. Both will have a huge impact on the story to come.
The Dream Stage is next in the tragedy treatment. During this stage, the protagonist becomes committed to his course-o-action, and it even looks like he'll win in the end.
For John and Cat's Cradle, that's the journey to San Lorenzo. Although John has no idea ice-nine is on the island, he still manages to do pretty well for himself. He becomes President, putting him in a position to do what is necessary to prevent tragedy. When ice-nine finally appears in "Papa" Monzano's corpse, John and the Hoenikkers do all they can to keep it from spreading.
Following the Dream Stage is the Frustration Stage, which can be, erm, frustrating for our characters. During this stage, things go insanely wrong for almost everyone involved, and the protagonist can no longer find respite or peace.
Care to guess when this stage hits? If you guessed the introduction of ice-nine into the ocean, then you've got it. Honestly, is there anything more frustrating than the end of the world? Well, perhaps surviving the end of the world….
Now time for the Nightmare Stage. It reads exactly as it sounds. John and his companions no longer have even the illusion of control over their fates. Many die of the ice-nine. Others commit suicide. The survivors have no idea what to do with the world presented to them. Those that do provide impractical solutions at best—such as Frank studying the ants, or Mrs. Crosby sewing the American flag.
And we conclude with the destruction stage, which is also exactly what it sounds like. Everyone, or nearly everyone, dies. Those that don't die may or may not learn a lesson about all the stuff that just happened to them.
John meets Bokonon, and the old religious figure suggests that he write a book on human stupidity and then commit suicide. Although we never see John's last act of rebellion, we can assume he chooses to follow Bokonon's suggestions. We never find out what happens to Newt or the Crosbys, but it seems safe to assume they're going to die.
Then again, who can't you say that about?