We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide (i) fundamental limits on the required number of broadcasts of data gathering, and (ii) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, \textcolor{black}{arbitrary geometric networks and random Erd$\ddot{o}$s-R$\acute{e}$nyi networks}, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight \textcolor{black}{in order sense}. Our main techniques are three distributed encoding techniques, called graph codes, which are designed respectively for the above-mentioned three scenarios. Our work thus extends and unifies previous works such as those of Gallager [1] and Karamchandani~\emph{et. al.} [2] on number of broadcasts for distributed function computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.