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In this paper we study the $L^p$ boundedness of the centred and the uncentredHardy--Littlewood maximal operators on the class $\Upsilon_{a,b}$, $2\leq a\leqb$, of trees with $(a,b)$-bounded geometry. We find the sharp range of $p$,depending on $a$ and $b$, where the centred maximal operator is bounded on$L^p(\mathfrak T)$ for all $\mathfrak T$ in $\Upsilon_{a,b}$. We show thatthere exists a tree in $\Upsilon_{a,b}$ for which the uncentred maximalfunction is bounded on $L^p$ if and only if $p=\infty$. We also extend theseresults to graphs which are strictly roughly isometric, in the sense of Kanai,to trees in the class $\Upsilon_{a,b}$.

Hardy-Littlewood maximal operators on trees with bounded geometry