An induced subgraph $G’$ of a graph $G$ is a subset of its vertices along with all the edges that are present in $G$ among those vertices. For $G’$ to be a tree, all vertices of a cycle in $G$ cannot be in $G’$. The tree composed of bold edges in the illustration given below shows an induced tree $T$. Vertex $u$ cannot be included in $T$ because u brings with itself edges $(u, v)$ and $(u, w)$ which completes a cycle $u-v-w-u$ and $T$ no longer remains a tree. $T$ is also a maximal induced tree because no more vertices can be included in $T$ without violating it being a tree.
The question is, what is a tight upper bound on the number of maximal induced trees in a connected triangulated planar graph.