How can I kill the swinging branches?

Hint 1 of 5:

Have you been underground yet?

Hint 2 of 5:

I'm talking about the area that you can reach via the giant cards.  See
another question in this section if you don't know how to get there.

Hint 3 of 5:

You'll need a weapon that's more suited for cutting down branches than
your fists or a gun.