Locally analytic representations of $p$-adic analytic groups have played a crucial role in many areas of arithmetic and representation theory (including in $p$-adic local Langlands program) since their introduction by Schneider and Teitelbaum. In this talk we will briefly review some aspects of the theory of locally analytic representations. Then, for a locally analytic representation $V$ of $GL_2(F)$ we will construct a coefficient system attached to the Bruhat-Tits tree of $Gl_2(F)$. Finally we will use this coefficient system to construct a resolution for locally analytic principal series of $GL_2(F)$.