Let T = (V, E) be a tree. We consider three colors (blue, green, red) and we want to assign a color to each vertex of the tree so that no two adjacent vertices have the same color. Precisely, a coloring is a function f : V → {blue, green, red} such that, for any u, v ∈ V, if u ∼ v then f(u) ≠ f(v). The picture below represents 3 different colorings of the same tree.
Let n ≥ 1 be an integer. Prove by induction that, if T has n vertices, then there are 3 ⋅ 2^{n-1} different colorings of T.