1. LetT= (V, E)be a rooted tree with an even number of vertices. Prove by induction that there exists a set of edges SE so that every vVis incident with an odd number of edges in S.
2. Hint for the inductive step: if the height of the tree ish >1choose an internal vertexuwith levelh1and children{u1,...,uk}. Try to delete a subset of{u,u1,...,uk}so that you can effectively apply induction.