A Tree Markov Decision Problem (T-MDP) is a finite-horizon MDP with a starting state $s_{1}$, in which every state is reachable from $s_{1}$ through exactly one state-action trajectory. T-MDPs arise naturally as abstractions of decision making in sequential games with perfect recall, against stationary opponents. We consider the problem of on-line learning in T-MDPs, both in the PAC and the regret-minimisation regimes. We show that well-known bandit algorithms -- \textsc{Lucb} and \textsc{Ucb} -- can be applied on T-MDPs by treating each policy as an arm. The apparent technical challenge in this approach is that the number of policies is exponential in the number of states. Our main innovation is in the design of confidence bounds based on data shared by the policies, so that the bandit algorithms can yet be implemented with polynomial memory and per-step computation. We obtain instance-dependent upper bounds on sample complexity and regret that sum a ``gap term'' from every terminal state, rather than every policy. Empirically, our algorithms consistently outperform available alternatives on a suite of hidden-information games.