Specializing Trees with Small Approximations I

Assuming $\mathrm{P}\mathrm{F}\mathrm{A}$, we shall use internally club ${\omega }_1$-guessing models as side conditions to show that for every tree $T$ of height ${\omega }_2$ without cofinal branches, there is a proper and ${\mathrm{\aleph }}_2$-preserving forcing notion with finite conditions which specialises $T$. Moreover, the forcing has the ${\omega }_1$-approximation property.