We consider multiple unmanned aerial vehicles (UAVs) serving a density of ground terminals (GTs) as base stations. The objective is to minimize the outage probability of GT-to-UAV transmissions. Optimal placement of UAVs under different UAV altitude constraints and GT densities is studied. First, using a random deployment argument, a general upper bound on the optimal outage probability is found for any density of GTs and any number of UAVs. A matching lower bound is also derived to show that the optimal outage probability decays exponentially with the number of UAVs. Next, the structure of optimal deployments is studied when the common altitude constraint is large. For a wide class of GT densities, it is shown that all UAVs should be placed to the same location in an optimal deployment. A design implication is that one can use a single multi-antenna UAV as opposed to multiple single-antenna UAVs without loss of optimality. This result is also extended to a practical variant of the Rician fading model recently developed by Azari et al. for UAV communications. Numerical deployment of UAVs in the centralized and practical distributed settings are carried out using the particle swarm optimization and modified gradient descent algorithms, respectively.