  In this work we consider Markov tree-shifts given by $k$ transition matrices,
one for each of its $k$ directions. We analyse some topological properties
introduced by arXiv:1509.01355 in order to answer some of the questions raised
by those authors. Moreover, we provide a method to characterize the complexity
function for Markov tree-shifts; this function is used to calculate the tree
entropies defined by arXiv:1712.02251 and arXiv:1509.08325. We compare both
entropies in order to determine some of its properties. Finally, the
characterization of the complexity function is used to calculate the entropy of
all binary Markov tree-shifts over the alphabet $\{0,1\}$.
