Two optimal leader selection problems are examined for multi-agent networks. The optimal leader set is the set of m > 0 leaders that maximizes performance of a linear dynamic network. In the problem for controllability, each leader is identified with a control input, and performance is measured by average controllability and reachable subspace volume. In the problem for robustness, each leader responds to an external signal, the linear dynamics are noisy, and the performance is measured by the steady-state system error. Previously, we showed that the optimal leader set for robustness maximizes a joint centrality in the network graph. In this paper, we show how the optimal leader set for controllability depends also on measures of the graph, including information centrality of leaders and eigenvectors of the graph Laplacian. We explore a fundamental trade-off between optimal leader selection for controllability and for robustness, and we outline a distributed algorithm for the selection of a pair of leaders in trees.