Generalized vertex-rankings of trees

We newly define a generalized vertex-ranking of a graph G as follows: for a positive integer c, a c-vertex-ranking of G is a labeling (ranking) of the vertices of G with integers such that, for any label i, every connected component of the graph obtained from G by deleting the vertices with label > i has at most c vertices with label i. Clearly an ordinary vertex-ranking is a l-vertex-ranking and vice-versa. We present an algorithm to find a c-vertex-ranking of a given tree T using the minimum number of ranks in time O(cn) where n is the number of vertices in T.