Why do terrain trees remain visible in complete darkness?

The shader for the tree leaves is Nature/Tree Soft Occlusion Leaves and the shader for the bark is Nature/Tree Soft Occlusion Bark. The only light in my scene comes from a spotlight on my player used as a flashlight, and the only thing visible outside the light are the trees. How do I get it so the trees are completely dark outside of the spotlight?

Does this help?