A k-cluster of a tree includes a single path and k-1 subpaths growing from that path. A k-MCore is a k-cluster that minimizes the sum of the distances of all vertices to the cluster plus the size of the cluster. This structure is motivated by the applications on overlay multicasting. The overlay multicast protocol constructs a virtual mesh spanning all member nodes of a multicast group. It employs standard unicast routing and forwarding to fulfill multicast functionality. In this paper, we propose effective distributed algorithms for constructing k-MCore on a tree network. The k-MCore is more stable and easier to maintain than the spanning tree in virtual mesh. The simulation results show that our approach handles the flexibility and mobility issues in an overlay multicast protocol effectively, especially when the group size is large.