Line jelly roll pan, 15-1/2x10-1/2x1", with waxed paper or baking parchment (I use the parchment). Grease (and I flour the bottom).
Mix flour, cocoa, baking powder, and salt; set aside.
Beat eggs in small mixer bowl (I do this in the big Kitchen Aid mixer with the wire beater) on high speed until very thick and lemon colored, 3-5 minutes.
Pour eggs into large mixer bowl if using a small one, and gradually beat in granulated sugar.
Beat in water and vanilla on low speed.
Gradually add flour mixture, beating just until batter is smooth.
Pour into pan, spreading batter to corners. Bake until wooden pick inserted in center comes out clean, 12-15 minutes.
Loosen cake from edges of pan; invert on towel generously sprinkled with powdered sugar. Carefully remove paper; trim stiff edges from cake if necessary.
While hot, roll cake and towel from narrow end.
Cool on a wire rack for 30-60 minutes.
Beat whipping cream, 2 tablespoons of sugar (more to suit taste), and the coffee, in chilled bowl until stiff.
Unroll cake carefully.
Spread whipped coffee cream filling over cake, reserving about 3-4 tablespoons in a small dish.
Re-roll, using towel to help, if necessary.
Cut a piece of the roll from one end, at an angle - maybe an inch at the narrow part to 2 inches at the wide part of the slice, and set aside for a moment while you begin to frost the log (Move the cake to it serving plate before you begin icing the log.) with the chocolate icing.
Make the icing by mixing the cocoa and margarine, then stirring in the powdered sugar, flavor, and hot water to a nice spreading consistency.
Set the waiting slice atop the frosted log - the icing should hold it in place sufficiently, or use a toothpick if you think you need to - to resemble a cut branch stub.
Ice those sides, too, blending into the main log part for a realistic look.
Use the reserved coffee whipped cream to cover the ends of the log and the top of the 'cut branch'.
Now's the fun: Using a fork -- I like one with tines rather apart -- draw the bark on the tree - just pull the fork through the soft frosting.
Rinse the fork and draw the circular rings on the ends (the coffee cream),