'A bird is sitting in the mango tree' is correct. Birds sit IN trees but ON branches (in trees)! The kind of tree (mango) goes before 'tree' - oak tree, apple tree, palm tree.

I agree with Claire, use "in". "Sitting on..." means the bird was sitting on top of the tree. Strange image. Perhaps the tree had fallen over? "Mango" is descriptive here, so "mango tree". "Tree of mango" could mean several strange things: maybe there is a person named Mango who owns the tree, or the tree is the most important tree in a land called Mango, or that the entire tree - trunk, branches, leaves, roots - is made out of mango fruit.

A bird is perching on the branch of a mango tree.