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This paper considers the problem of selecting a low stretch spanning tree for a given planar graph. The stretch of the tree T over a vertex pair u,w is defined as the ratio between the distance from u to w in T and their distance in the original graph. Two central problems that arise regarding the worst-case and average stretch are (1) the minimum stretch spanning tree (MSST) problem, introduced in [CC95], which asks for finding a spanning tree minimizing the maximum stretch over all vertex pairs, and (2) the minimum average stretch spanning tree (MAST) problem, introduced in [AKPW95]. Dealing with t-spanner trees, it is shown in [FK98] that even for unweighted planar graph it is NP-hard to determine the optimal maximum stretch. This paper focuses on the MSST and MAST problems on unweighted planar graphs. It is shown that both problems can be solved exactly in polynomial time on outerplanar graphs, and also in the special case of 1-face depth graphs in which no interior vertex has degree 2. Then, a 2k-approximation algorithm is proposed for the MAST problem on general planar graphs, where k is the face depth of the graph.