Consider the following 2-respecting min-cut problem. Given a weighted graph and its spanning tree , find the minimum cut among the cuts that contain at most two edges in . This problem is an important subroutine in Karger's celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs.