Is this tree minimization problem NP-complete?
Your problem is ill-defined. Since the functions $f\colon \mathcal{T} \rightarrow \mathbb{Q}$ and $\mu\colon 2^N \rightarrow \mathbb{Q}$ (rather than real-valued) have a very long description, you can go over all possible trees $T \in \mathcal{T}$ in polynomial time.

To make the question more meaningful you might want the algorithm to have oracle access to $f$ and $\mu$, and then instead of NP-completeness you might want to prove a lower bound on the number of evaluations of $f$ and $\mu$.

Another possibility is that $f$ and $\mu$ have some structure and so can be specified succinctly. If this is the case, then you need to refine the problem reflecting that feature.

A similar problem occurs in submodular optimization. Suppose we want to maximize a monotone submodular function over a matroid constraint. There are several randomized polynomial time algorithms giving a $1-1/e$ approximation. There is a lower bound by Vondrák showing that in the (so-called) value oracle model, any better approximation requires evaluating the function at exponentially many places (this lower bound is information-theoretic rather than complexity-theoretic). Feige proved that a special case of the problem, maximum coverage, is NP-hard to approximate better than this ratio. Finally, Vondrák and Dobzinski show how to lift the value oracle result to a complexity result in the succinct representation model, assuming RP$\neq$NP (rather than P$\neq$NP as in Feige's result); the technique (whose main ingredient is list-decoding) also gives results that Feige's can't.