Triangular maximal operators on locally finite trees

We introduce the centred and the uncentred triangular maximal operators tau and U, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both tau and U are bounded on L-p for every p in (1, infinity], that tau is also bounded on L-1(I), and that U is not of weak type (1, 1) on homogeneous trees. Our proof of the L-p boundedness of U hinges on the geometric approach of Cordoba and Fefferman. We also establish L-p bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy-Littlewood maximal operators (on balls) may be unbounded on L-p for every p < infinity even on some trees where the number of neighbours is uniformly bounded.