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 ∈ T : Mf(x) > λ}) ≤ cs
1
λ
Z
T
|f(x)|M(w
s
)(x)
1
s dx 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 [38] is established.
