A k-tree is a graph that can be formed by starting with the complete graph K_k and iterating the operation of making a new vertex adjacent to all the vertices of a k-clique of the existing graph. For order n > k + 1, a k-path graph is a k-tree with exactly two vertices of degree k. We develop a formula for the number of unlabeled k-paths of order n. In particular, there is a bijection between these graphs and equivalence classes of strings of numbers from {1,2,...,k} under a relation that treats them as equivalent when they can be made the same by permutation of their numbers and possible reversal of the string.