Abstract: A class of measures is considered on homogeneous trees whose distance to the origin is exponentially decreasing. The definitions of Lebesgue spaces, BMO (bounded mean oscillation) spaces, maximal operators and their commutators for this type of measure on homogeneous trees are given. By using the decomposition theory of homogeneous trees, the boundedness of maximal operators and their commutators in Lebesgue spaces and some equivalent properties are proved.

Key words: homogeneous tree, maximal operator, commutator, exponential decline measure