“Mango” or “mango tree”?

Solution 1:

Me? I say "apple tree" for the plant, and "apple" for the fruit. Likewise for mango, lemon, coconut, walnut, etc.

But if we're only talking about a particular variety of apple then the term "tree" may be omitted, but it's not a given, e.g. "cox's orange pippin"

Cox’s Orange Pippin Apple has been an English institution for over 150 years. It arose as a chance seedling in the 1820s in the orchard of a retired brewer named Richard Cox, near the present-day location of Heathrow Airport. It’s a yellow apple with a red blush, which makes it appear orange—hence, the “Orange” part of its name. The “Pippin” part of its name is an old English word for an Apple Tree grown from a seed, or “pip.” Cox’s Orange Pippin wasn’t made available to other growers until the 1850s, and it wasn’t grown commercially until the 1860s, but once it finally became known, it won the hearts of a nation. (source)

Oak, Ash, Birch, Beech, Pine, Sycamore, and Willow, etc. are well-known trees, we don't normally speak about their fruits, so actually saying "oak tree" is a bit redundant.

Below are images of different trees taken from Wikipedia, note that when the fruit is commonly known, the noun "tree" is needed as in “lemon tree” and “walnut tree” whereas “Beech” and “Cypress” do not require it.

Solution 2:

Obviously, when referring to a tree which has the same name as a fruit or other product obtained from it, you need to make it clear that you mean the tree and not the product. "I leaned my back against an oak" (a line from a folk song) is not ambiguous, but "I leaned against an apple" would be. However, if you were pointing to trees in an orchard you could perfectly well say "That's an apple and that's a pear".

Solution 3:

Most plants have a principal use. Its noun in isolation refers to that principal implied use. If it is not meant that way then a specific description is added to refer to its other attribute.

Mango refers to the fruit and mango tree or mango leaf are other non principal descriptors. If mango is cut it is in a kitchen for eating rather than an outdoor activity of cutting a branch of it using an axe.

Oak, Pine refer to their timber and if a landscape is referred to, it mentions oak tree or pine cone.

Whiskey is the alcoholic liquid drink but we refer to whiskey bottle, whiskey smell when referring to these in particular.

Tea or coffee are the beverages but we can have tea/coffee aroma or tea/coffee plantation or trade.