We consider the following fragmentation model of the unit interval $\QTR{Bbb}{I}$: we start fragmenting $\QTR{Bbb}{I}$ into two pieces with uniform random sizes. One of these two sub-intervals is then chosen at random according to a $\beta -$ size-biased picking procedure, $\beta \in \QTR{Bbb}{R}.$ This particular tagged fragment is next broken into two random pieces, one of which is chosen accordingly at random for further splitting; the process is then iterated independently. This model constitutes a recursive $\beta -$partitioning procedure where splitting occurs only at one of the two fragments which were formed in the previous step of fragmentation. We investigate some statistical features of this model.