All Arbitrarily Fixed Generalized Banana Trees Are Graceful

Consider a set of caterpillars, having equal and fixed diameter, in which one of the penultimate vertices is of arbitrary degree and all the other internal vertices including the other penultimate vertex are of fixed even degree. Merge an end-vertex adjacent to the penultimate vertex of fixed even degree of each of such caterpillars together. The rooted tree thus obtained is called Arbitrarily Fixed Generalized Banana Tree. In this paper we prove that all arbitrarily fixed generalized banana trees are graceful. This would imply that “all banana trees are graceful” and “all generalized banana trees are graceful” as corollaries.