Games are a medium for interaction and storytelling. While both of these elements often go hand in hand, there are times in which the story crafted by the developer and the player's personal story (as told by their actions and experiences) differ widely. Our medium is great at creating interactive systems and letting players loose within them, yet the stories we tell in our games often fall back onto the tropes, conventions, and static structures found in books or movies.
While it's possible to implement heavily interactive storylines with dozens of branching paths, and even systems that create and assemble the story procedurally, the same development effort is much better placed with core gameplay mechanics. After all, everyone will interact with gameplay systems, but only a small minority will explore all story branches. This is why in most cases, our games can provide a lot more freedom in their gameplay than in their storytelling.
Let's look at the possible structures...