We model the growth of a cell population using a piecewise deterministic Markov branching tree. In this model, each cell splits into two offspring at a division rate $B(x)$, which depends on its size $x$. The size of each cell increases exponentially over time, with a growth rate that varies for each individual. Expanding upon the model studied in \cite{hof}, we introduce a scenario with two types of bacteria: those with a young pole and those with an old pole. Additionally, we account for the possibility that a bacterium may not always divide into exactly two offspring. We will demonstrate that our branching process is well-defined and that it satisfies a many-to-one formula. Furthermore, we establish that the mean empirical measure of the model adheres to a growth-fragmentation equation when structured by size, growth rate, and type as state variables.