Summary: We consider a polling network with a single server that serves the queues in a cyclic order. Upon the completion of his service at queue \(i\), a customer is routed to queue \(j\) with probability \(p_{i, j}\), or leaves the system with probability \(p_{i, 0}\). The service policy is branching-type. Using the limit theorem of a multi-type branching process with immigration, we derive the asymptotic behavior of joint queue lengths at polling epochs in the heavy traffic. With the help of heavy traffic averaging principle, the limit behavior of the mean waiting times is also presented.