  We propose an adaptive estimator for the stationary distribution of a
bifurcating Markov Chain on $\mathbb R^d$. Bifurcating Markov chains (BMC for
short) are a class of stochastic processes indexed by regular binary trees. A
kernel estimator is proposed whose bandwidth is selected by a method inspired
by the works of Goldenshluger and Lepski [18]. Drawing inspiration from
dimension jump methods for model selection, we also provide an algorithm to
select the best constant in the penalty.
