Given a tree $T$ (with $n \ge 2$ vertices). Suppose we have written $0$ or $1$ on each vertex of $T$ such that the sum of all numbers is an even number. Prove that it is possible to choose a subset $F$ of the edges of $T$ such that every vertex with label $0$ is adjacent to an even number of edges of $F$ and every vertex with label $1$ is adjacent to an odd number of edges of $F$