In spanning tree optimization of graphs, there are two important criteria, namely, dilation (stretch) and congestion. This gives rise to two combinatorial optimization problems as follows. The minimum stretch spanning tree problem is to find a spanning tree of a graph such that the maximum distance of all embedded edges in the tree is minimized. The minimum congestion spanning tree problem is to find a spanning tree of a graph such that the maximum number of overlapped edges for all embedded edges in the tree is minimized. These problems bring us two graph-theoretic parameters, tree stretch and tree congestion. In this paper we mainly study the structural properties of tree stretch and tree congestion, including the duality of fundamental cycle and fundamental cocycle, extremal property, lower and upper bounds, optimality characterization, and optimal value computation.