We extend the classical notion of block structure for periodic orbits of interval maps to the setting of tree maps and study the algebraic properties of the Markov matrix of a periodic tree pattern having a block structure. We also prove a formula which relates the topological entropy of a pattern having a block structure with that of the underlying periodic pattern obtained by collapsing each block to a point, and characterize the structure of the zero entropy patterns in terms of block structures. Finally, we prove that an n-periodic pattern has zero (positive) entropy if and only if all n-periodic patterns obtained by considering the k\mathrm{th} iterate of the map on the invariant set have zero (respectively, positive) entropy, for each k relatively prime to n

The authors have been partially supported by MEC grant numbers MTM2008-01486 and MTM2011-26995-C02-01