  We study properties of the boundary trace operator on the Sobolev space
$W^1_1(\Omega)$. Using the density result by Koskela and Zhang, we define a
surjective operator \mbox{$Tr: W^1_1(\Omega_K)\rightarrow X(\Omega_K)$}, where
$\Omega_K$ is von Koch's snowflake and $X(\Omega_K)$ is a trace space with the
quotient norm. Since $\Omega_K$ is a uniform domain whose boundary is
Ahlfors-regular with an exponent strictly bigger than one, it was shown by L.
Mal\'y that there exists a right inverse to $Tr$, i.e. a linear operator $S:
X(\Omega_K) \rightarrow W^1_1(\Omega_K)$ such that $Tr \circ S=
Id_{X(\Omega_K)}$. In this paper we provide a different, purely combinatorial
proof based on geometrical structure of von Koch's snowflake. Moreover we
identify the isomorphism class of the trace space as $\ell_1$. As an additional
consequence of our approach we obtain a simple proof of the Peetre's theorem
about non-existence of the right inverse for domain $\Omega$ with regular
boundary, which explains Banach space geometry cause for this phenomenon.
