Lets go one by one,
(i) A graph in which there is a unique path between every pair of vertices is a tree.
==> This statement is true, Because graph can have unique path only when it does not have cycle. And according to the definition of tree, its a graph without cycel. Hence this is a valid statement.
(ii) A connected graph with e = v − 1 is a tree.
==> This statement is true. Not every graph with e = v - 1, will be a tree. But if the graph is connected hence its true.
(iii) A graph with e = v − 1 that has no circuit is a tree.
==> This statement is also true, Here he has not mentioned the connected thing, but mentioned that it has no circuit. It means that its connected.
You can also proof these, just by taking some example. For the formal Proof check this.