These are two big classic questions that have haunted every branding expert for ages. The first one is about “possibility” of branding anything; the second one is about “appropriateness” of branding everything around us.
To answer these questions from the fundamental ground of branding, we need to focus on defining a brand in the first place. While there are numerous definitions, let us choose the most simple yet the comprehensive one. Branding has necessarily two dimensions that even all debating scholars will agree upon: (i) identification, and (ii) significant differences from others within and outside its native category. Evidently, everything around us, including a banana tree in your backyard that is not branded, has these two characteristics in it. Therefore, anything can be branded. Now it is upon you whether you would promote this as a brand or not.
Here comes the second question. Should everything be branded? Should that banana tree in your backyard be branded? From business point of view, you can brand a banana tree, provided the benefit outweighs the cost. Does that tree have any historical significance for which people will pay to visit your place and see the tree? Does it have any special feature for which it will generate revenue? So everything does not need to be branded. Here is a word of caution: having no branding is like having no face, and having “no face” is a type of identity and differentiation that leads to some sort branding! A brand is what it does. Therefore, even if you do not go for promoting an active brand, whatever you deal with will gradually take over an image synonymous to a “brand”. Therefore, even though not everything needs to be branded, it would be wise to see (and manage) everything by keeping the concept of branding in mind.