Preheat the oven to 200ºC / gas mark 6. Lightly grease a 33 × 23 centimetre Swiss roll tin, and line with non-stick paper or baking parchment, pushing it into the corners.
For the sponge, whisk the eggs and sugar using an electric hand whisk in a large bowl until the mixture is pale in colour, light and frothy. Sift the flour and cocoa powder into the bowl and carefully cut and fold together, using a metal spoon, until all the cocoa and flour are incorporated into the egg mixture. (Be careful not to beat any of the air out of the mixture.)
Pour into the lined tin and spread evenly out into the corners.
Bake in the middle of the preheated oven for eight to 10 minutes until pale golden and the sides are shrinking away from the edge of the tin.
Place a piece of baking parchment bigger than the Swiss roll on the work surface. Invert the cake on to the paper and remove the bottom lining piece of paper.
Trim the edges of the cake with a sharp knife and make a score mark 2.5 centimetre in along the longer edge. Roll up (from the longer edge) using the paper, rolling with the paper inside. Set aside to cool.
While the cake is cooling, make the icing. Melt the chocolate and 300ml of the cream in a bowl over a pan of simmering water until completely melted (be careful not to overheat; the bowl must not touch the water). Put into the fridge to cool and firm up (this icing needs to be very thick for piping). Whip the remaining cream.
Uncurl the cold Swiss roll and remove the paper. Spread a third of the icing over the surface, then spread the whipped cream on top, and re-roll tightly. Cut a quarter of the cake off from one end on the diagonal. Transfer the large piece of cake to a serving plate and angle the cut end to the side of the large cake to make a branch. Cover the surface of the cake with the melted apricot jam.
Put the remaining chocolate icing into a piping bag fitted with a star nozzle. Pipe long thick lines along the cake, covering the cake completely so it looks like the bark of a tree. Cover each end with icing or, if you wish to see the cream, leave uniced.