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Median linear regression modeling is a natural approach for analyzing censored survival or failure time data. A median linear model lends itself to a simple interpretation that is particularly suitable for making direct predictions of survival or failure times. We propose a new, unique, and efficient algorithm for tree-structured median regression modeling that combines the merits of both a median regression model and a tree-structured model. We propose and discuss loss functions for constructing this tree-structured median model and investigate their effects on the determination of tree size. We also propose a split covariate selection algorithm by using residual analysis (RA) rather than loss function reduction. The RA approach allows for the selection of the correct split covariate fairly well, regardless of the distribution of covariates. The loss function with the transformed data performs well in comparison to that with raw or uncensored data in determining the right tree size. Unlike other survival trees, the proposed median regression tree is useful in directly predicting survival or failure times for partitioned homogeneous patient groups as well as in revealing and interpreting complex covariate structures. Furthermore, a median regression tree can be easily generalized to a quantile regression tree with a user-chosen quantile between 0 and 1. We have demonstrated the proposed method with two real data sets and have compared the results with existing regression trees.