Abstract: In this paper, we
show that the graceful tree conjecture is true for a new family of rooted trees.
More precisely, we prove that the following theorem.Theorem:
Rooted trees obtained by identifying one of the end-vertices adjacent to either
of the penultimate vertices of any number of caterpillars having equal diameter with the property that all the
internal vertices of each of such caterpillars are exclusively of odd degrees,
are graceful.
