Asynchronous task allocation is a fundamental problem in distributed computing, in which p asyn- chronous processes must execute a set of m tasks. Also known as write-all or do-all, this problem been studied extensively, both independently and as a key building block for various distributed algorithms. In this paper, we break new ground on this classic problem: we introduce the To-DoTree concurrent data structure, which improves on the best known random- ized and deterministic upper bounds. In the presence of an adaptive adversary, the randomized To-DoTree algorithm has O(m+plogplog2m) work complexity. We then show that there exists a deterministic vari- ant of the To-DoTree algorithm with work complexity O(m+p log5 m log2 max(m, p)). For all values of m and p, our algorithms are within log factors of the Ω(m + p log p) lower bound for this problem. The key technical ingredient in our results is a new approach for analyzing concurrent executions against a strong adaptive scheduler. This technique allows us to handle the complex dependencies between the processes’ coin flips and their scheduling, and to tightly bound the work needed to perform subsets of the tasks. We believe this technique will be useful in the analysis of other, more complex, concurrent data structures.