2. (12 marks)
Let T be a rooted tree. Let $V_0$ be the set of vertices whose level is even, and $V_1$ the set
of vertices whose level is odd. Prove each of the following statements:
(a) The following two equations both hold:
$V_0 \cup V_1 = V$,
$V_0 \cap V_1 = \emptyset$.
(b) For any two distinct vertices $x$ and $y$ in $T$ whose levels are equal mod 2, they cannot
be adjacent.
(c) If $T$ has height 1 then $T$ is the complete bipartite graph $K_{|V_0|, |V_1|}$
