This paper proposes a distributed borrowing addressing (DIBA) scheme to solve problems of failure in address assignments resulting from limited tree depth and width when the distributed address assignment mechanism is used in a ZigBee/IEEE 802.15.4 wireless sensor network. DIBA is a method of borrowing addresses from neighbor nodes for newly entering nodes and assigning the borrowed addresses. Its network or sensing coverage can increase with almost the same overhead as the existing method. DIBA is a simple and lightweight means of addressing and routing, making it suitable for wireless sensor networks. Simulations showed that DIBA is a distributed addressing scheme with consistently excellent performance.