I followed all of the instructions to get a golden shovel but why does the sapling of the tree keep dying

If you plant the tree too close to another tree, there are too many weeds around, or possibly it is the wrong season (I'm not sure about the last one), then the sapling might die. Otherwise you may have a glitch. I only have animal crossing: population growing for the gamecube, so it may be different in other versions of animal crossing, but we (my family and friends) have always been able to get the golden shovel, and the sapling always continues to grow after you harvest the shovel.