Accurate skeleton segmentation of the entire anteroposterior bone scintigrams of the human body is essential for diagnosing bone metastases. However, conventional methods lack a loss design incorporating prior anatomical information, leading to segmentation failures, particularly when dealing with the irregular shapes of organs or high concentrations of positive accumulation. Cases where diagnostic support systems present anatomically abnormal findings may shatter the confidence of doctors and their reliability in these systems. In this paper, we propose a novel multi-factor component tree loss function to resolve the topological issues in segmentation failures. The proposed loss function, computed based on the component trees, comprises two factors: image maxima vanishment and reconnection. We aim to discard the false positive connected components (FPCCs) and reconnect the disconnected true positive connected components (TPCCs) for each bone. Experiments conducted on a private bone scintigrams dataset show that our proposed method outperforms state-of-the-art approaches in dice similarity coefficient (DSC) while efficiently addressing topological issues at a low computational cost. Code is available at https://github.com/MultiCTree/MultiCTree.