Consider the following inhomogeneous fragmentation model: suppose an initial particle with mass $x_{0}\in $ $\left( 0,1\right) $ undergoes splitting into $b>1$ fragments of random sizes with some size-dependent probability $p\left( x_{0}\right) $. With probability $1-p\left( x_{0}\right) $, this fragment is left unchanged for ever. Iterate the splitting procedure on each sub-fragment if any, independently. Two cases are considered: the stable (unstable) case with $p\left( x_{0}\right) =x_{0}^{a}$ (respectively $p\left( x_{0}\right) =1-x_{0}^{a}$), for some $a>0.$ In the first (second) case, smaller fragments splitting with smaller (larger) probability, one suspects some stabilization (instability) of the fragmentation process. Some statistical features are studied in each case, chiefly fragments size distribution, partition function, structure of the underlying random fragmentation tree.
