1. Preheat oven to 325 degrees. Brush two 9-by-2-inch round cake pans with butter; line bottoms with parchment. Butter parchment. Dust with flour, tapping out excess.

2. Whisk together flour, baking powder, and salt in a medium bowl. In the bowl of a mixer fitted with the paddle attachment, beat butter with coconut oil on medium speed until light and fluffy, about 3 minutes. Add confectioners' sugar; beat on low speed until combined, then increase speed to medium and beat until light and fluffy, 2 to 3 minutes more. Add egg whites and vanilla; beat until combined, about 1 minute.

3. Reduce speed to low; beat in flour mixture in 3 batches, alternating with coconut milk in 2 batches, until just combined.

4. Divide batter between prepared pans; smooth tops. Bake until a tester inserted in centers comes out clean, 35 to 40 minutes. Let cool in pans set on a wire rack 5 minutes. Turn out onto rack; let cool completely.

5. With a serrated knife, trim tops of cakes level. In a bowl, fold shredded coconut into 1 cup buttercream. Spread evenly on one cake layer. Top with second cake layer, trimmed-side down. Reserve 2 cups buttercream; use remainder to spread a smooth layer over entire cake.

6. Place reserved buttercream in a pastry bag fitted with a large star tip (such as Ateco #826). Use a tree-shaped cookie cutter to mark an overlapping pattern on top of cake, then fill in each tree by piping within markings: Working from bottom to top, hold pastry bag at a 45-degree angle and place tip on cake; squeeze while pulling forward and then up, releasing pressure to make pointed tips. Cake can be stored in refrigerator in an airtight container up to 1 day; bring to room temperature before serving.