Connor is very agile and flexible - he can easily climb trees. Below you can find a description of this mechanic.
A new addition to Assassin's Creed III are big and vast forests. Moving between trees is very similar to navigating cities, but features a few special principles. Basically, during free run, Connor will always move horizontally through the forest. When running along a branch he will come across a trunk, he will avoid it and move to the branch on the other side.
If you want to climb higher, you need to use the jump button. The same button can be used when standing between two branches to make Connor climb higher.
Forest climbing paths usually begin with fallen trees. From them you can easily reach upper parts of the woods.
Another new addition are complex rock formations which you can climb by using narrow ledges and wide, vertical cracks.