We propose a new self-stabilizing anonymous leader election algorithm in a tree graph. We show the correctness of the protocol and show that the protocol terminates in O(n⁴) time starting from any arbitrary initial state. The protocol can elect either a leaf node or a non leaf node (starting from the same initial state) depending on the behavior of the daemon. The protocol generalizes the result in [A self-stabilizing leader election algorithm for tree graphs] and at the same time is much simpler and terminates in polynomial number of moves.