The $d$ dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can be solved optimaly in polynomial time (Chung, [1984]).
So, what about Trees in higher (fixed) dimension? To be more precise:
$d$-dimensional Arrangement Problem of Trees:
$\text{Input}: d$, a Tree $T=(V,E),$ $|V|=n=a^d$
$\text{Task}$: Find a bijection $b: V\rightarrow\{{1,\ldots,\sqrt[d]{n}\}}^d$, such that
$$\sum_{e=(v,w)\in E}|b(v)-b(w)|_1$$ is minimized.
Can we generalize an Algorithm from the linear case, or can we show that this problem is still $NP$-hard?