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We study the abelian sandpile model on decorated one dimensional chains. Wedetermine the structure and the asymptotic form of distribution ofavalanche-sizes in these models, and show that these differ qualitatively fromthe behavior on a simple linear chain. We find that the probabilitydistribution 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 linearcombination of two simple scaling forms $Prob_L(s) = 1/L f_1(s/L) + 1/L^2f_2(s/L^2)$, for large $L$, where $f_1$ and $f_2$ are some scaling functions ofone argument.Comment: 10 pages, revtex, figures include

Breakdown of simple scaling in Abelian sandpile models in one dimension