Graph Theory: Is this a correct statement?

Question

I'm asked to prove this statement in my assignment:

Suppose that $T$ is a tree. Prove that degree of every vertex of $T$ is odd if and only if for each $e \in E(t)$, size of both components of $T-e$ are odd.

Is this correct? Shouldn't size of each component be even?

Edit: Seems size of each components refers to number of vertices and not edges, therefore it's correct :)

Answer 1

No, the problem seems right as worded. Notice that it's talking about node counts, not edge counts. To take an example, look at a tree on four vertices such that one root and the other three are the root's children. This has odd degree for every vertex. Deleting any one edge gives two connected components: one vertex by itself and the other three together.

Answer 2

Yes. For instance, look at the tree with two vertices and one edge.