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You store the current depth of every branch for each node, and append Rendertree to render all nodes up to a specific depth, given as a parameter.

This way, you can show each depth level seperatly.

Pseudocode could be like this:

Generate tree

While(GameLoopRunning)
  begin
    for i := 0 to maxdepth do
      begin
        RenderTree(trunk, i)
        delay(250); //should not be like this, will block entire game loop.         
      end
  end

Edit: You might want to use some kind of timer to fire some kind of GrowNow!-Events to your game logic in order to avoid the for-loop and delay in your main game loop.