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 and a weight , the objective is to enumerate all weighted trees that share the same tree structure as and have a total weight of . 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 vertices and a total weight of for any given positive inputs and .