West of Fortree past the bridge.
The Weather Institute is not in Heart Gold. It is in Ruby, Sapphire, and Emerald.
The Weather Institute is located to the left of Fortree City.
beat the grunts
There is not a weather institute in Pokemon Heartgold, you are thinking of Pokemon ruby, sapphire, and emerald, there is not a weather institute in Pokemon heartgold.
it is left of fortree city
A very low chance like 1 in 100
after you beat the the bad people in the weather institute in ruby emerald or sapphire one of the good people will give you a castform.
To get the leaf stone in Pokémon Ruby you need to first go to the area of the Weather Institute. Next surf north to the waterfall and you will find the stone.
You need to beat the elite four then talk to the man on the computer at the Weather institute. He will tell you what route groudon's cave is on.
Simply defeat your rival after beating Team Magma/Aqua at the Weather Institute on Route 119. She/he will give you HM02 (Fly) after you defeat them.
You have to play through the plot of the story. After you save the Weather Institute and leave there, your rival will appear and battle you, and give you Fly when you win.
Sorry to say but there is no weather institute in Pokemon soulsilver version