Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Functional programming, design and analysis of algorithms / 6 / d

No. A tree consisting of a single edge $uv$ has the two maximum independent sets $\{u\}$ and $\{v\}$. Each has size one, and neither can be enlarged because $u,v$ are adjacent. Thus even a two-vertex tree gives a counterexample to uniqueness.