A definition of the Tree Property can be found in The "tree property" and dividing types.
In Shelah's Classification theory there is a proof of the equivalence between the $k$-Tree Property and the 2-Tree Property but I find it a bit convoluted. I remember reading somewhere a direct proof of this fact but I can't find it or prove it myself.
The idea I was trying is to start with a witness for the $k$-TP (i.e., some formula $\varphi$ and some tree $(a_\eta)_{\eta\in 2^{<\omega}})$ and modify it somehow, for example changing $\varphi$ for a conjunction of $k-2$ instances of $\varphi$. This guarantees the 2-inconsistent condition but we may loose consistency over any branch.
Any hint or reference would be appreciated.