I think that in all classical models of TP($\omega_2$) we have $2^{\omega_0}=\omega_2$. Is there a known model of TP($\omega_2$) + $2^{\omega_0}>\omega_2$ at all?
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I think Spencer Unger in "Fragility and indestructibility of the tree property" proves that if one adds an arbitrary number of Cohen reals to Mitchell's model, then the tree property survives (see http://www.math.cmu.edu/~sunger/). 

