Wood floats or sinks because of its density, not because of its weight. Domestic woods float when completely dry, while some other woods do not float even when dry. Wood float because its cell lumens (the insides of the cells) are empty and contracted. When you put dry wood onto water it floats, but then it starts to take on water, the cells fill and expand and it eventually sinks.
It might feel heavy, but most kinds of it are lighter than water in terms of density – otherwise it wouldn’t be able to float. If you look at the first link in citations, it lists the densities of wood from a lot of different trees. You can see that some trees produce wood that is actually more dense, such as ebony and lignum vitae; on the other hand, balsa wood is 7-9 times lighter than water.
For comparison, second link lists water density at slightly below 1000 kg/m^3 (slightly below 0.0624 lb/ft3) at room temperature.
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