Constant time enumeration of weighted trees

This study specifically addresses the enumeration problem of the class of weighted trees. A weighted tree is a rooted unordered tree, and each vertex in a weighted tree is assigned an integer weight. The task is to enumerate the weighted trees. Given a tree T and a weight w , the objective is to enumerate all weighted trees that share the same tree structure as T and have a total weight of w . Note that the weight assigned to each vertex must be non-negative. The algorithm utilizes reverse search and enumerates each weighted tree in a constant amortized time. By combining our findings with the enumeration of the class of rooted trees, we can efficiently enumerate every weighted tree with a maximum of n vertices and a total weight of w for any given positive inputs n and w .