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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 $$\begin{align*}& w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}\: \text{d}x\qquad s>1\end{align*}$$ is settled, and moreover, it is shown that it is sharp, in the sense that it does not hold in general if |$s=1$|⁠. Some examples of nontrivial 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. [ABSTRACT FROM AUTHOR]