Let (G, w) be a weighted graph with the adjacency matrix A. The group inverse of (G, w), denoted by \((G^{\#},w^{\#})\) is the weighted graph with the weight \(w^{\#}(v_iv_j)\) of an edge \(v_iv_j\) in \(G^{\#}\) is defined as the ijth entry of \(A^{\#}\) , the group inverse of A. We study the group inverse of singular weighted trees. It is shown that if (T, w) is a singular weighted tree, then \((T^{\#},w^{\#})\) is again a weighted tree if and only if (T, w) is a star tree, which in turn holds if and only if \((T^{\#},w^{\#})\) is graph isomorphic to (T, w). A new class \(\mathbb {T}_w\) of weighted trees is introduced and studied here. It is shown that the group inverse of the adjacency matrix of a positively weighted tree in \(\mathbb {T}_w\) is signature similar to a non-negative matrix.
