In this article, we develop a theory of maximal functions associated with natural approach regions in the setting of spaces of homogeneous type. We construct a natural approach family \(\mathcal {G}\) and an approach family \(\mathcal {L}\) such that \(\mathcal {L}\) satisfies the \(\mathcal {G}\) -tent condition, and then define maximal operators \(M_{\mathcal {G}}\) and \(M_{\mathcal {L},\mathcal {G}}\) . The \(L^p\) boundedness, Fefferman-Stein inequalities, and characterizations of weighted weak \(L^1\) boundedness for both operators \(M_{\mathcal {G}}\) and \(M_{\mathcal {L},\mathcal {G}}\) are established. Furthermore, for an infinite tree T endowed with a very regular property, we construct a space of homogeneous type \((bT, \mu _{{T}}, d_e)\) , where bT is the boundary of T, \(\mu _{{T}}\) is the hitting distribution and \(d_e\) is a metric on T. We define the maximal operator \(\mathcal {H}_{\mathcal {L}}\) on the space \((bT, \mu _{{T}})\) , and show the \(L^p\) boundedness, Fefferman-Stein inequality, and characterization of weighted weak \(L^1\) boundedness for \(\mathcal {H}_{\mathcal {L}}.\)
