Popular Monte-Carlo tree search (MCTS) algorithms for online planning, such as epsilon-greedy tree search and UCT, aim at rapidly identifying a reasonably good action, but provide rather poor worst-case guarantees on performance improvement over time. In contrast, a recently introduced MCTS algorithm BRUE guarantees exponential-rate improvement over time, yet it is not geared towards identifying reasonably good choices right at the go. We take a stand on the individual strengths of these two classes of algorithms, and show how they can be effectively connected. We then rationalize a principle of "selective tree expansion", and suggest a concrete implementation of this principle within MCTS. The resulting algorithm,s favorably compete with other MCTS algorithms under short planning times, while preserving the attractive convergence properties of BRUE.