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In 1975, Leech introduced the problem of finding trees whose edges can be labeled with positive integers in such a way that the set of distances (sums of weights) between vertices is \(\{1, 2, \dots, \binom{n}{2}\}\), where \(n\) is the number of vertices. We refer to such trees as perfect distance trees. More generally, we define a distinct distance tree to be a weighted tree in which the distances between vertices are distinct. In this article, we focus on identifying minimal distinct distance trees. These are the distinct distance trees on \(n\) vertices that minimize the maximum distance between vertices. We determine \(M(n)\), the maximum distance in a minimal distinct distance tree on \(n\) vertices, for \(n \leq 10\), and give bounds on \(M(n)\) for \(n \geq 11\). This includes a determination of all perfect distance trees for \(n < 18\). We then consider trees according to their diameter and show that there are no further perfect distance trees with diameter at most \(3\). Finally, generalizations to graphs, forests, and distinct distance sets are considered.