Tree Unattractiveness

You are given a tree with $N$ vertices (numbered $1$ through $N$) and a bag with $N$ markers. There is an integer written on each marker; each of these integers is $0$, $1$ or $2$. You must assign exactly one marker to each vertex. Let's define the *unattractiveness* of the resulting tree as the maximum absolute difference of integers written on the markers in any two vertices which are connect