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As a cost-efficient scalable Internet-access network, we have studied the architecture, protocols, and design optimizations of the Wireless Internet-access Mesh NETwork (WIMNET). WIMNET is composed of wirelessly connected multiple access-points (APs), where at least one AP acts as a Gateway (GW) to the Internet. Two types of APs are used for the hierarchical configuration to achieve scalability with sufficient bandwidth and low costs, namely, a programmable Smart AP (SAP) equipped with multiple antennas, and a non-programmable Conventional AP (CAP) with a single antenna. For a SAP, a smart antenna using an array antenna should be adopted to enhance the performance by allowing simultaneous activations of multiple links. Because the transmission capacity of a SAP with m antennas can be ideally m times of that of a CAP, the routing path should consider this difference so that the average delay between the GW and any AP be minimized to avoid bottlenecks. In this paper, we present an extension of our previous routing tree algorithm for this smart antenna use in WIMNET to improve the throughput by considering the feature. We evaluate the throughput improvement through simulations using the WIMNET simulator.