  We study the abelian sandpile model on decorated one dimensional chains. We
determine the structure and the asymptotic form of distribution of
avalanche-sizes in these models, and show that these differ qualitatively from
the behavior on a simple linear chain. We find that the probability
distribution of the total number of topplings $s$ on a finite system of size
$L$ is not described by a simple finite size scaling form, but by a linear
combination of two simple scaling forms $Prob_L(s) = 1/L f_1(s/L) + 1/L^2
f_2(s/L^2)$, for large $L$, where $f_1$ and $f_2$ are some scaling functions of
one argument.
