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The growing demand for online media content delivery and multi-player gaming is expected to increase the amount of multicast service requests in both public and private networks. Careful traffic engineering of multicast service requests is becoming increasingly essential, as establishing the lowest cost tree, e.g., shortest path tree, in the network for every individual multicast request does not always ensure a global optimization. In this paper, the authors investigate the use of alternate tree routing, according to which multiple sub-optimal tree candidates are computed for each multicast request. When performing global routing optimization, each request is then assigned one of the tree candidates, in a way that yields the desired result, e.g., to minimize the number of (active) links that are required in the network to bear the entire set of requests. A key component of the investigated approach is the timely construction of the set of alternate trees for every given root and destination set. An algorithm is proposed to compute multiple sub-optimal tree candidates with a guaranteed upper bound on the maximum end-to-end transmission latency. The algorithm builds on the widely used K shortest path algorithm, generalizing the technique of computing K candidate shortest paths to obtain a number of candidate sub-optimal trees with desirable properties. Simulation experiments obtained for a multi-protocol label switching (MPLS) traffic engineering network are discussed to investigate the effectiveness, performance, and scalability of the proposed algorithm.