Surprised and frightened, you totter weakly toward the chair by the fire, but you trip on the sleeping dog. You fall toward the fire, hitting your head on the stone fireplace. You lie dazed on the floor, vaguely realizing that your clothes have caught fire, and the fire has spread to the chair and rug. You try to rise but it is too late; smoke overcomes you and you die in pain.

Return to the beginning of the story.

This is too soon to end a branch of the story tree. 

This branch of the story is small in order that it won't get in the way of our meta-story.

But some of the branches don't really develop a plot. A four-node path like this is only an episode or vignette. The problem is that our slimmed down demo story barely qualifies as a narrative.

There is little time for a beginning, middle, and end, which Aristotle says is necessary for a narrative. But in branching stories some branches need to be short. A friend who wrote several of the old Choose Your Own Adventure books said:

The problem with the Choose books is there isn't much there there. Each individual story is no more than five or six brief scenes long . . . like a six panel cartoon. That's not enough. But if they tried to make the stories longer, they would quickly get into unwieldiness. It's sort of like the mathmatical example of the guy who starts with a penny income a day but doubles the income every day and becomes a millionaire by the end of the month.

Then why call them stories at all?

Because the ongoing narrative here is not the set of nodes but the travail of the reader, who is telling him or herself a meta-narrative about exploring the nodes and being blocked and beginning again. That narrative does have a beginning, a puzzling middle, and a triumphant ending.

That is one of the performative stories you mentioned.

It has been planned out by the writer. But what makes it different from a linear prose story is that the writer can only design the landscape, not control the reader's choice of paths.

A hypertext story that is free from the restrictions of tree structure, such as Afternoon [9] or Victory Garden [19], can provide more complex landscapes and paths, and more intense narrative experience of suspense and connection, and so more readerly adventure in exploring.

Yes, but even in our little story tree there are still the implicit meta-stories. And the examples you cited don't just keep expanding. Although their link structure is much more complex than a tree they do have something like a convergence and senses of completion to their narrative landscape. There's no endless exfoliation of story paths.

We model the threat of that elsewhere in our little story.