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We consider the model of Brownian motion indexed by the Brownian tree, whichhas appeared in a variety of different contexts in probability, statisticalphysics and combinatorics. For this model, the total occupation measure isknown to have a continuously differentiable density. Although the densityprocess indexed by nonnegative reals is not Markov, we prove that the pairconsisting of the density and its derivative is a time-homogeneous Markovprocess. We also establish a similar result for the local times ofone-dimensional super-Brownian motion. Our methods rely on the excursion theoryfor Brownian motion indexed by the Brownian tree.

The Markov property of local times of Brownian motion indexed by the Brownian tree