Of a place: having many trees.
"The first thing he said was he was 'fraid she'd got into the woods. You know they always find um in the woods in the papers. They git confused 'n' they wander round." / "Yes," said Mr. Townshend, "that's about it, you see. I thought right away of Noah's Island; it's the woodsiest place there is about. I sh'll git some men together right away.[…]"
From woods + -y (suffix meaning ‘having the quality of’ forming adjectives), to distinguish the word from woody (“made of wood, or having wood-like properties, etc.”).
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