OED pergola: an archway in a garden or park consisting of a framework covered with trained climbing or trailing plants.
Does 'trained' here means that the plant has been 'trained' to grow in a specific path? If so, I didn't know trained could be applied to anything besides people or animals.
Yes, you can train a plant to grow a certain way. It involves staking or tying a branch in a particular way that makes it grow where you want it to. There are also various tricks in topiary and bonsai to train plants to grow in unusual shapes and sizes.
Naturally, "train" means something slightly different for plants, because you have to physically force them to do something, rather than persuading them to. Also, plants tend not to remember what they've been trained to do.