A new model of causal failure is presented and used to solve a novel replica placement problem in data centers. The model describes dependencies among system components as a directed graph. A replica placement is defined as a subset of vertices in such a graph. A criterion for optimizing replica placements is formalized and explained. In this work, the optimization goal is to avoid choosing placements in which a single failure event is likely to wipe out multiple replicas. Using this criterion, a fast algorithm is given for the scenario in which the dependency model is a tree. The main contribution of the paper is an $O(n + ρ\log ρ)$ dynamic programming algorithm for placing $ρ$ replicas on a tree with $n$ vertices. This algorithm exhibits the interesting property that only two subproblems need to be recursively considered at each stage. An $O(n^2 ρ)$ greedy algorithm is also briefly reported.