The p-median problem on general networks has been studied since the 1960s. Kariv and Hakimi [10] showed that this problem is NP-hard even if the network is a planar graph of maximum degree 3. In the case of tree networks the p-median problem is solvable in polynomial time. Kariv and Hakimi [10] developed an algorithm that computes a solution in O(p 2 n 2) time. The running time was improved to O(pn 2) by Tamir [16] and later to O(n lg p+2 n) by Benkoczi and Bhattacharya [2]. Better bounds are known for the special cases (on trees) where p = 1 or 2. Goldman [6] gave an O(n) algorithm for the 1-median problem on trees. The 2-median problem was studied by Mirchandani and Oudjit [11], whose localization properties were later used to improve the O(n 2) bound (derived from the general tree case) to O(n lg n)-see papers by Hämäläinen [9] and Gavish and Sridhar [5]. We present a framework for solving the 2-median problem on trees, building on earlier work. Our framework leads to several algorithms with o(n lg n) runtime, i.e., better than the current best-known O(n lg n) runtime, in common special cases. The time bounds are: (i) O(n lg wmax/ lg n), where wmax is the largest largest node weight, which is linear when node weights are bounded by a polynomial in n; (ii) O(n lg nL), where nL is the number of leaves in in the tree; (iii) O(ndmax), where dmax is the maximum edge length, which is linear when edge lengths are bounded by a constant; and (iv) O(n lg), where is the number of nodes on the trunk, an easily identified path that is guaranteed to contain at least one of the two medians.