Single-path routing is widely used in wireless networks due to low resource consumption. However, it is vulnerable to link failure because such a failure may adversely affect an entire path. To overcome this, multipath routing has been proposed providing fault-tolerance. In this paper, we propose a novel multipath routing protocol called traffic-aware stateless multipath routing (TSMR) based on an overlaid tree topology comprising two topologies, namely, bounded degree tree (BDT) and root-oriented directional tree (RODT). BDT is strong on reducing routing overhead, and RODT is resilient against lossy links. By synergistically overlaying them, TSMR dynamically selects the local optimal path according to the given traffic flow and the failure on the primary path. In particular, TSMR enables stateless and low overhead routing despite multipath routing by keeping only one-hop neighbors to maintain multiple paths. To evaluate TSMR, we conducted simulations with a shadowing model reflecting lossy links, and compared with single and multipath routing protocols, such as ZTR, STR, AODV, and RPL. The simulation results show that the overall performance of TSMR surpasses that of others for packet delivery ratio, end-to-end delay, control overhead, memory consumption, and power consumption regardless of network size, number of sessions, and traffic flow.