From infinitely many supercompacts, tree property at ℵ_n (2 ≤ n < ω) is consistent with arbitrary continuum function below ℵ_ω obeying the basic inequality 2^{ℵ_n} > ℵ_{n+1}.
Cardinal invariants above the continuum
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Easton's theorem for the tree property below aleph_omega
From infinitely many supercompacts, tree property at ℵ_n (2 ≤ n < ω) is consistent with arbitrary continuum function below ℵ_ω obeying the basic inequality 2^{ℵ_n} > ℵ_{n+1}.