  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 the following Fefferman-Stein estimate \[ w\left(\left\{ x\in
T\,:\,Mf(x)>\lambda\right\} \right)\leq
c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 \] 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 on infinite trees is established.
