  Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance
between the vertices $u$ and $v$ in $G$. A set
$S=\{s_{1},s_{2},\cdots,s_{n}\}\subset V(G)$ is said to be a {\em resolving
set} if for all distinct vertices $u,v$ of $G$, there exist an element $s\in S$
such that $d(s,u)\neq d(s,v)$. The minimum cardinality of a resolving set for a
graph $G$ is called the {\em metric dimension} of $G$ and it is denoted by
$\beta{(G)}$. A resolving set having $\beta{(G)}$ number of vertices is named
as {\em metric basis} of $G$. The metric dimension problem is to find a metric
basis in a graph $G$, and it has several real-life applications in network
theory, telecommunication, image processing, pattern recognition, and many
other fields. In this article, we consider {\em cube of trees} $T^{3}=(V, E)$,
where any two vertices $u,v$ are adjacent if and only if the distance between
them is less than equal to three in $T$. We establish the necessary and
sufficient conditions of a vertex subset of $V$ to become a resolving set for
$T^{3}$. This helps determine the tight bounds (upper and lower) for the metric
dimension of $T^{3}$. Then, for certain well-known cubes of trees, such as
caterpillars, lobsters, spiders, and $d$-regular trees, we establish the
boundaries of the metric dimension. Further, we characterize some restricted
families of cube of trees satisfying $\beta{(T^{3})}=\beta{(T)}$. We provide a
construction showing the existence of a cube of tree attaining every positive
integer value as their metric dimension.
