  For a graph property $\Pi$, Subgraph Complementation to $\Pi$ is the problem
to find whether there is a subset $S$ of vertices of the input graph $G$ such
that modifying $G$ by complementing the subgraph induced by $S$ results in a
graph satisfying the property $\Pi$. We prove that the problem of Subgraph
Complementation to $T$-free graphs is NP-Complete, for $T$ being a tree, except
for 41 trees of at most 13 vertices (a graph is $T$-free if it does not contain
any induced copies of $T$). This result, along with the 4 known polynomial-time
solvable cases (when $T$ is a path on at most 4 vertices), leaves behind 37
open cases. Further, we prove that these hard problems do not admit any
subexponential-time algorithms, assuming the Exponential Time Hypothesis. As an
additional result, we obtain that Subgraph Complementation to paw-free graphs
can be solved in polynomial-time.
