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A set of edges \(D\) in a graph \(G\) is a dominating set of edges if every edge not in \(D\) is adjacent to at least one edge in \(D\). The minimum cardinality of an edge dominating set of \(G\) is the edge domination number of \(G\), denoted \(D_E(G)\). A graph \(G\) is edge domination critical, or \(EDC\), if for any vertex \(v\) in \(G\) we have \(D_E(G – v) = D_E(G) – 1\). Every graph \(G\) must have an induced subgraph \(F\) such that \(F\) is \(EDC\) and \(D_E(G) = D_E(F)\). In this paper, we prove that no tree with more than 2 vertices is \(EDC\), develop a forbidden subgraph characterization for the edge domination number of a tree, and we develop a construction that conserves the \(EDC\) property.