If your computer were unable to handle materials, your image would be all black.

Unlit does only affect the influence of static and dynamic lights. It does not affect the sun or any other entity shading parameters. For disabling the shading you have to set the material's diffuse color vector at zero.

I'm afraid there is no way around reading the manual about materials - "My computer can't handle materials" is no excuse, unless you inherited your computer from your grandfather.

By the way, your tree sprites look nice.