Let T = ( V, E) be an undirected tree with positive edge lengths. Let S be a subset of V with k. For each vertex v let d 1 v ≦…≦ d k v be the sorted sequence of distances from v to the k vertices in S. For i = 1,…, k, let S( v, i) denote the vertex set of the minimal subtree containing v and the vertices of S whose distance from v is at most d i v . For r>0 let N( v, r) denote the set of vertices in V whose distance from v is at most r. We prove that the collection of all subsets { S( v, i)∩ N( v, r)}, with v in V, i = 1,…, k, and d i−1 v ≦ r≦ d i v , ( d 0 v = 0), is totally balanced. We also show that the subcollection of all subsets { S( v, i)} with v in V is totally unimodular. These results extend and unify some previous results on collections of subtrees of a tree, and they imply the existence of polynomial algorithms for several location models on trees. Finally we discuss extensions of the above results to bitrees and strongly chordal graphs.