Minimum edge cover with a constraint
There is at most one edge cover which touches each vertex an odd number of times. To see this, let us consider a more general problem, in which each vertex is labeled either "odd" or "even". We want to cover each vertex by an odd or even number of edges, according to its label.

If the tree contains more than one vertex, then we can always find some leaf. It it is labeled "odd", every solution must contain the unique edge containing it. We add this edge to the edge cover, remove the leaf, and modify the label at the other end of the edge accordingly. If the leaf is labeled "even", every solution must not contain the unique edge containing it, and so we simply remove the vertex. Eventually only one vertex will remain. There is a solution if and only if this vertex is labeled "even".