r \right\}$ denote the ball of radius $r$ centered at $x$. Let $X$ be a maximal set of vertices of $G$ that have pairwise distance at least $2m+1$ from each other. Thus, for distinct $x,y\in X$, the balls $B_m(x)$ and $B_m(y)$ are disjoint. Extend the $(B_m(x))_{x\in X}$ to disjoint connected subgraphs of $G$ by first adding each vertex at distance $m+1$ from $X$ to one of the $B_m(x)$ to which it is adjacent. Then add each vertex at distance $m+2$ from $X$ to one of the subgraphs constructed in the previous step to which it is adjacent. Continue like this until each vertex of $G$ is contained in one of the constructed subgraphs and denote the subgraph obtained from $B_m(x)$ in this way by $T(x)$. The choice of $X$ implies that each vertex of $G$ has distance at most $2m$ from $X$, so each vertex of $T(x)$ has distance at most $2m$ from $x$ in $T(x)$. Therefore, as $G$ has girth at least $4m+2$, each $T(x)$ is an induced subtree of $G$. In particular, $B_m(x)$ is a tree in which every vertex that is not a leaf has degree at least $d$ and in which every leaf has distance $m$ 