  Suppose $\varphi$ is a holomorphic self map of the unit disk and $C_\varphi$
is a composition operator with symbol $\varphi$ that fixes the origin and
$0<|\varphi'(0)|<1$. This work explores sufficient conditions that ensure all
holomorphic solutions of Schr\"oder equation for the composition operator
$C_\varphi$ belong to a Bloch-type space $\mathcal{B}_\alpha$ for some
$\alpha>0$. The results from composition operators have been extended to
weighted composition operators in the second part of this work.
