Given a graph $G$ of $n$ nodes partitioned into facilities and customers, the $r$-edge interdiction covering problem (REIC) is to remove up to $r$ edges so as to maximize the total weight of customers disconnected from all facilities, which is called the covering objective function. While REIC is known to be NP-complete for general graphs, Fröhlich and Ruzika show that the problem can be solved in polynomial time when $G$ is a tree, providing an $O(n^7 r)$-time algorithm. We give an efficient $O(nr^2)$-time dynamic programming algorithm for REIC on trees that is fixed-parameter linear in $n$. Evaluating our solution on a benchmark of randomly generated tree networks with baselines of the Fröhlich and Ruzika algorithm and the Gurobi integer program solver, we demonstrate that in practice, our algorithm is both significantly faster and less sensitive to network topology and size. We extend our algorithm for REIC to graphs of bounded treewidth, a well-studied family of sparse graphs that generalizes trees, and obtain a matching runtime of $O(nr^2)$. We also consider the $r$-facility interdiction covering problem (RFIC), a novel variant of this network interdiction problem where the goal is to remove up to $r$ facilities to maximize the covering objective function over disconnected customers. We show that RFIC is NP-complete by observing it generalizes the small set bipartite vertex expansion problem (SSBVE), also known as the minimum $p$-union problem. We give an $O(nr^2)$-time algorithm for RFIC on trees, which also gives an $O(n^3)$-time algorithm for SSBVE on trees.