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The diffusion of rumors on microblogs generally follows a propagation tree structure, that provides valuable clues on how an original message is transmitted and responded by users over time. Recent studies reveal that rumor detection and stance detection are two different but relevant tasks which can jointly enhance each other, e.g., rumors can be debunked by cross-checking the stances conveyed by their relevant microblog posts, and stances are also conditioned on the nature of the rumor. However, most stance detection methods require enormous post-level stance labels for training, which are labor-intensive given a large number of posts. Enlightened by Multiple Instance Learning (MIL) scheme, we first represent the diffusion of claims with bottom-up and top-down trees, then propose two tree-structured weakly supervised frameworks to jointly classify rumors and stances, where only the bag-level labels concerning claim's veracity are needed. Specifically, we convert the multi-class problem into a multiple MIL-based binary classification problem where each binary model focuses on differentiating a target stance or rumor type and other types. Finally, we propose a hierarchical attention mechanism to aggregate the binary predictions, including (1) a bottom-up or top-down tree attention layer to aggregate binary stances into binary veracity; and (2) a discriminative attention layer to aggregate the binary class into finer-grained classes. Extensive experiments conducted on three Twitter-based datasets demonstrate promising performance of our model on both claim-level rumor detection and post-level stance classification compared with state-of-the-art methods.