The color cost of a caterpillar

Using the positive integers as colors, the cost of a given coloring of the nodes of a graph G is the sum of its colors. The color cost of G is then the smallest cost of any proper coloring of G, We specify three types of caterpillar using their codes. These specifications enable the representation of an arbitrary caterpillar as a sequence of these types. The representation is utilized to develop a fast algorithm for calculating the color cost of a given caterpillar.