We study the properties of Schnyders realizers and canonical ordering trees of plane graphs. Based on these newly discovered properties, we obtain compact drawings of two styles for any plane graph G with n vertices. First, we show that G has a visibility representation with height at most 15n/16 . This improves the previous best bound of (n - 1). Second, we show that every plane graph G has a straight-line grid embedding on an (n - ₀ - 1) × (n - ₀ - 1) grid, where ₀ is the number of cyclic faces of G with respect to its minimum realizer. This improves the previous best bound of (n - 1) × (n - 1). We also study the properties of the regular edge labeling of 4-connected plane triangulation. Based on these properties, we show that every such a graph has a canonical ordering tree with at most (n + 1)/2 leaves. This improves the previously known bound of (2n + 1)/3 . We show that every 4-connected plane graph has a visibility representation with height at most 3n/4 . All drawings discussed in this paper can be obtained in linear time.