One of the wireless mesh network's important features is each node can support more than one transmission rate. However, few previous literatures on the broadcast tree construction take this into account. Recently, some researchers proposed to reduce the network wide broadcast transmission latency by taking advantage of the multi-rate nature. However, it suffers from a long construction time as analyzed in this paper, which brings in a long start-up delay. This paper proposes a fast broadcast tree construction algorithm (called Rate First) by exploiting the relationship between the transmission rate and its range. Simulation results show that it does not only keep the broadcast transmission latency at the same level with the stateof-the-art work, but also accomplishes in a significantly short time.