This article investigates the controllability problem of multi-agent systems. Each agent is assumed to be goveed by a second-order consensus control law corresponding to a directed and weighted graph. Two types of topology are considered. The fi rst is conceed with directed trees, which represent the class of topology with minimum information exchange among all controllable topologies. A very simple necessary and suffcient condition regarding the weighting scheme is obtained for the controllability of double integrator multi-agent systems in this scenario. The second is conceed with a more general graph that can be reduced to a directed tree by contracting a cluster of nodes to a component. A similar necessary and suffcient condition is derived. Finally, several illustrative examples are provided to demonstrate the theoretical analysis results.