And feel free to rewrite this to make it clearer. You can certainly have a graph in b), the naive implementation is a tree.

With persistent levels, either a) the each 'floor' of the game is always the same once generated and all exits from the current level lead to one of two adjacent levels, or b) each exit from a level leads to a new level, which results in the game space consisting of a tree of levels branching away from the root which the player first entered where the number of children of the level equals the number of exits from that level, minus the one that the player arrived from.

Why is it only the case in a) that "each 'floor' of the game is always the same once generated". I would think this by definition is always true for persistent levels.

In b) I'm confused if it's meant that in the tree of levels, the player can go up to the 'parent' of the current level (i.e. the level where he came from) or not. If he can't, and a given level is only ever entered once, I'd say it's a case of non-persistent levels - unless they're the same the next time the game is played from scratch. If he can, then surely the parent levels should be the same as before in order for this to qualify as persistent levels?

Also, I would like to rewrite the paragraph to say that levels can be connected by arbitrary graphs rather than saying they must be connected either in sequence or in a tree structure, which is an arbitrary limitation of the definition in my opinion.