In this paper weighted endpoint estimates for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree are provided. Motivated by Naor and Tao [23] the following Fefferman-Stein estimate $$ w({ x\in T:Mf(x)>\lambda})\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 $$ is settled and moreover it is shown it is sharp, in the sense that it does not hold in general if $s=1$. Some examples of non trivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A strong Fefferman-Stein type estimate and as a consequence some vector valued extensions are obtained. In the Appendix a weighted counterpart of the abstract theorem of Soria and Tradacete [38] on infinite trees is established.