The Mangrove Tree is specific to the Don't Starve: Shipwrecked DLC Content.
It is found growing in the waters of the Mangrove biome, and can only be accessed by boat or raft. It has a distinctive appearance with a bulbous trunk and roots that begin above the waterline.
When chopped down, the player receives 2 logs and 1 stick, 2 logs and 2 stick or 3 logs and 3 stick.
As of now, there is no way to replant Mangrove Trees, and the player does not get anything special by chopping them down, so there is no point in doing so (unless the player is desperate for sticks and can't find any Saplings.)