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In this paper, we develop a random access scheme which combines the widely used binary exponential backoff (BEB) algorithm with a cross-layer tree algorithm (TA) that relies on successive interference cancellation (SIC) with first success (FS). BEB and SICTA/FS complement each other nicely in enabling the novel protocol to attain a maximum stable throughput (MST) as high as 0.6 without packet loss. Although BEB-SICTA/FS avoids the deadlock problem caused by the error propagation commonly present in successive interference cancellation (SIC) algorithms, it may still suffer from deadlock effects induced by the "level skipping" caused by harsh wireless fading effects. We further develop a novel BEB-SICTA/F1 protocol, which is a modified version of BEB-SICTA/FS. Analysis and simulations demonstrate that this simple modification leads to high-throughput random access while completely avoiding deadlock problems.