How do you cut down the little tree on Pokemon heart gold?

You can't actually cut down the little tree in Pokémon HeartGold. It's a decorative element and serves no purpose other than being part of the environment.

There are a few tools that can affect trees in other Pokémon games, like the Chop Chop move in Pokémon Ruby, Sapphire, and Emerald, or the Cut HM in games like FireRed and LeafGreen. However, these options are not available in HeartGold.