The borrowed address algorithm can solve the orphan problem made by the restriction of threshold of children when DAAM(distributed address assignment mechanism) assigns the addresses for the nodes in ZigBee networks. The exited borrowed algorithms only can increase the success ratio of address assignment, but their overhead and times spend on founding net cost much. To address the problem, we propose an efficient borrowed address assignment algorithm EDAA BA (efficient distributed address assignment algorithm based on borrowed address). To control the overhead and time spend on founding net, it borrows addresses firstly from the descent nodes in same branch for the orphan router nodes, adds the mechanism of immediate reply of borrowed message, and assigns the remainder addresses unused by DAAM in the 16 bits addresses for the orphan end devices. Theoretical and simulation analyses show that our algorithm outperforms DAAM and two of its present improvement algorithms in terms of the overhead and time spent on forming network, under the premise of better success rate of address assignment.