In the example below, an infiltrating force of veterans are scouting a potential target for a later raid. If the scouting goes well, they can then attempt to sabotage the target. If it doesn't go well, then the enemy has prepared an ambush for the veterans:
The results of the scout scenario determine what's happening in the narrative and over a few lines, you can develop quite a varied outcome to a campaign. With the second example, we've expanded the levels to a 3 game campaign. It starts with the original two levels of the previous example, with a scout scenario leading to either a sabotage or an ambush scenario:
With the third game, there are four possibilities depending on the results of the previous scenario. Doing well in the sabotage scenario leads on to an assassination mission as a secondary objective; failure means trying to evacuate. From the ambush scenario, a rearguard action follows a successful breakout from the ambush whereas it comes down to a final stand if you fail in the ambush. This simple 3 level tree campaign gives you a great variety in games and has a narrative. The narrative goes from a glorious mission where things were sabotaged and generals killed to a complete nightmare when the forces are spotted scouting, ambushed, and then slaughtered in a final stand.
One final thing about our last example: if you notice, a win for the initial attackers means taking the left hand path down the tree whereas a loss means taking the right hand path. Where you reach from left to right at the bottom of the tree gives you an idea of how well (narratively speaking) the attackers did overall in the campaign. This is a simple way of scoring the campaign using the narrative. Alternatively, you can let the winner of the scenario decide which one to play next. This adds a small amount of strategy to the campaign with players picking scenarios that best suit their forces.
Tree campaigns are a little more demanding in their planning than ladder campaigns, with you needing to devise scenarios that you might not play but they offer so much more in the way of the campaign’s narrative. It’s also worth noting that whilst tree and ladder campaigns may be about the narrative, that doesn't mean they don't have to be competitive as well: there's a number of different ways to score a narrative campaign.
I hope I explained that clearly enough, if you've got anything to say or ask, use the comments below.