You can try using contradiction.
For example, take an arbitrary rooted tree with at least two vertices. You need to show that there exists at least 1 non-terminal vertex with said property. So for contradiction, you can assume that there is NO such vertex, i.e. there isn't any non-terminal vertex, all of whose children are terminal. An equivalent statement is: for each non-terminal vertex, one or more children are non-terminal. From this, I think, you can get a contradiction very easily.