I'm a tad confused as to how this problem is posed. A tree has precisely one spanning subgraph (itself).

What also confuses me, although less so, is that the result isn't actually true. Take 4 vertices and place them in a line, so you have a tree with 4 vertices and 3 edges. Clearly such a tree does not have a spanning tree where each vertex has odd degree.

So, I'm not entirely sure how to help you. I would point out, however, that if you have an even number of vertices then there exists a tree which spans these vertices, which is simply the tree with one (unique) vertex connected to all the others. Therefore, if there are 2n vertices this vertex will have degree 2n-1, while all the other vertices have degree 1. This may or may not be the result you are looking for...