We consider diffusion on a tree with nodes that randomly switch between allowing and prohibiting particles to pass. We find exact expressions for various splitting probabilities and mean first passage times for a single diffusing particle and show how the many parameters in the problem, such as the node gating statistics and tree topology, contribute to these exit statistics. We also consider a concentration of particles that can always pass through interior branch nodes and determine how an intermittent source at one end of the tree affects the flux at the other end. The latter problem is motivated by applications to insect respiration.