Final answer:
Every tree, including rooted Fibonacci trees, can be colored with two colors by coloring the root one color and alternating colors for each level of adjacent vertices. This is due to the absence of cycles in trees, which ensures no coloring conflicts arise.
Step-by-step explanation:
Every tree can be colored using only two colors, a characteristic that is relevant to graph theory. A tree is a connected graph with no cycles, meaning that there is exactly one path between any two vertices. The rooted Fibonacci trees, denoted as Tn, are a specific type of tree constructed recursively: Ti and T2 are the initial trees with a single vertex, and Tn for n ≥ 3 is formed by connecting a new root vertex to trees Tn-1 and Tn-2, respectively, as the left and right subtrees.
To color a tree using two colors, one could use a basic algorithm. Start by coloring the root vertex one color (say, red), and then color every child the opposite color (say, green). Because a tree does not contain cycles, this algorithm will not produce any conflicts. Therefore, after traversing the entire tree, every vertex will be colored, and each vertex will be adjacent only to vertices of the opposite color, ensuring a proper two-coloring.
This principle can be applied to rooted Fibonacci trees as well. Since Tn has Tn-1 and Tn-2 as subtrees, they can be separately colored using two colors. It follows that the entire tree Tn can be colored correctly. Because the two subtrees are joined only via the root of Tn and the subtree roots are of different colors, there will be no conflict. Therefore, rooted Fibonacci trees confirm the property that all trees are two-colorable.