Odd-graceful labelings of trees of diameter 5

A difference vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy the weight |f(x)-f(y)|. A difference vertex labeling f of a graph G of size n is odd-graceful if f is an injection from V(G) to {0,1,...,2n-1} such that the induced weights are {1,3,...,2n-1}. We show here that any forest whose components are caterpillars is odd-graceful. We also show that every tree of diameter up to five is odd-graceful.