Limited - selective with a series of yes/no or black/white objectives in a game and the outcome/choices that the player chooses will branch out to the most appropriate thread. 

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Open-Ended - player is able to freely travel through the world and have the ability to choose where and when to visit particular places in the game. this includes 'sandbox' which gives players the freedom to interact and modify with the world however they want.

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Funneling Narrative (Chokepoints) - this is fairly common because you have a controllable and definable way to set the player back on the narrative arc of the game. You also give the player more freedom to explore and you can determine when and where you will allow the story progression to take place.

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For example, you can allow the players to explore the environment and multiple story threads, but the player will not be able to progress further into the game until the player goes to the checkpoint e.g a bartender in town. The bartender becomes a game play and a story chokepoint which allows the player further game play.

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Chokepoints in this type of structure tend to be the story set pieces for the game.

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Critical Paths - this structure has one success path which allows the player to deviate from it in a very small way. However, nothing of consequence that happen in the game or story happens outside of the predetermined path that winds its way through the experience.

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Nodal Storytelling - this tells stories of the location and/or the objective dependent. Each node is a self-contained piece with a setup, midpoint, and payoff. Each of these story nodes may lead to a larger reveal or it may lead to something that the player may like to see on the journey throughout the game.

Usually, the story structure doesn't function as a traditional branch (and it may/may not be considered valid), but since moving from node to node of a story it is often dependent of a previous element of a story. 

nodal structure can be considered as a pseudo-branch.

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Linear - opposite of open-ended and these types of games use scripted events, cut scenes, and restricted paths to control the story so that players can see how the writer wanted the story to be.

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Non-Linear - story starts with a flashback or at the end and the rest of the game consists of present events leading to that final event which was shown at the start. the beginning, middle, and end are not in an expected order.