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This paper focuses on the problem of increasing the traffic capacity (volume of admissible traffic) of broadcast and multicast flows in a wireless mesh network (WMN). We study and suggest routing strategies where the process of constructing the forwarding tree considers three distinct features: (a) the ability of individual mesh nodes to perform link-layer broadcasts at multiple rates, (b) the wireless broadcast advantage, whereby a single broadcast transmission covers multiple neighboring receivers and (c) the residual transmission capacity at a WMN node, subject to intereference-based constraints from existing traffic flows in its neighborhood. Our metric of interest is the the total number of broadcast and multicast flows that can be admitted into the network, without resulting in unacceptable degradation in metrics such as packet loss and dissemination latency. Our discrete event simulations show that the broadcast tree construction heuristic which takes both transmission rate and residual bandwidth into account out-performs those that do not. Building on our work on resource-aware broadcast tree construction, we propose a resource-aware multicast tree construction algorithm which exploits the multiple link-layer rates, the wireless broadcast advantage and the amount of resources available. Simulation results show that this algorithm performs better than heuristics based on pruning a broadcast tree or shortest path trees.