"How do we know that the deforestation occurred because of the statue erecting; perhaps they just used a lot of firewood?"

When the Dutch discovered Easter Island the cult of the heads had finished and the cult of the Birdman was in place. The people of the Island told the Dutch about the demise of the head cult and why the Birdman cult was started. The story is really fascinating, but too long to go into here. There's a great book about it all, the title of which escapes me ...

Anyway, the Dutch records show that the people of Easter Island (killed off by VD and other diseases brought to them by westerners) told that they deforrested the Island to make sledges, trackways and towers to erect the figures.