Wood floats, after all. Rowing boats just swamp - they don’t sink I think. Of course, if it’s laden with cannon, balls, swords, muskets and ammo that would make it sink. A merchant ship full of pottery or metal ingots or sundry heavy stuff would no doubt sink. By the way, I’m thinking of big old wooden ships from before steel hulls and before copper bottoms. There would be very little (expensive) metal compared to the weight of wood. Furniture, containers, all wood. Heaver than water stuff - nails, hinges, crockery, cutlery and glassware for the officers, some bits of tackle. Sails float. Hemp rope floats, I think. I just read that before plimsole lines British owners would send ships out so full of cargo the main deck was only 3 feet above the water. If the ship made it, fine. If it sank, also fine because they were fully insured. And this was when press ganging was the only way to recruit a crew, merchant or Royal Navy. A merchant sailor couldn’t resign for years and jumping ship was a capital offence. Those ship owners make Victorian factory owners seem virtuous. Anyway, a lot of ships must have sunk. Lots of cargo, like tea or dry goods, is no heavier than water, and after unloading many ships would sail without much or even any cargo, no matter how efficient owners were at keeping 'em loaded, and sink in heavy seas. But I haven’t heard that there used to be hundreds of ships wallowing up to the gunwhales around the oceans.
In order to stay right side up, ships carry a lot of ballast in thier bildges. More than enough to offset the bouyancy of the material from which thier hull was constructed.
In some cases this could be replaced by cargo, others not.
A very wide ship doesn’t need such ballast, which makes it slow, and unable to handle high seas. One class of such ships is known as barges.
Another such class is known as landing craft. Thanks, Kevbo.
On a wooden monohull (or one of fiberglass, etc.), ballast is placed in the bottom of the ship or in the keel to keep the up side up. Consequently, when the ship swamps and fills with water, it is now heavier than the water it displaces - this is the definition of having a “sinking feeling”. 
Smaller sailing boats may have a hull design that generates a sufficient righting moment when heeled over that they don’t need ballast - these boats might float when capsized. This is also the case for catamarans. Larger vessels might carry enough flotation material that they could float when swamped - but not merchant vessels, since all that volume would presumably be filled with cargo.
Hope that helps.
Wood is generally bouyant but not all that bouyant. (I know, it depends on the type of wood, it’s growth density, etc., etc.) A log in the water floats but it’s waterline isn’t very low on the log.
When they were doing a lot of logging around Lake Superior they would float the logs in the lake for transport. Some of the logs would sink. Because of the huge scale of logging, the small percentage of logs that sunk were enough that they are now being “mined”. It seems that due to the way that the cold, fresh water of the lake preserved for them for 100 years, they are a source of prime wood that is exceptional for specialty purposes like musical instruments.
My point, which I should have added to the last post, is that it is not really the wood that is keeping the ship afloat, it is the water displacement due to the hull design.
Here is a list of wood that doesn’t float, with specific gravity listed:
Lignum vitae (Guaicum officinale, 1.37)
Quebracho (Schinopsis balansae, 1.28)
Pau d’arco (Tabebuia serratifolia, 1.20)
Knob-thorn (Acacia pallens, 1.19)
Desert ironwood (Olneya tesota, 1.15)
Ebony (Diospyros ebenum, 1.12)
As to why ships made of lighter wood sink. The bouyancy of the ship is determined not by the wood, but by the design of the ship. The ship is designed so that the water presses the weight of the ship upwards. If the ship either falls over too far, or the design is altered, say by holes in the hull, the ship loses this bouyancy.
In this case the ship will float, provided the weight of the ship is equal to or less than the bouyancy of the component parts. Unfortunately, most of the hardwoods used to build sturdy wood ships are not all that inherently bouyant. Thus, the rigging and any trapped load cause the ship to sink.
Not by any means a complete list. For a start, many eucalypts produce wood that doesn’t float.
Some wooden boats don’t sink. I’ve seen Indonesian fishing vessels that were completely swamped with just the top of the mast sticking out of the water. They can stay like that for weeks.
The answer is that, yes, much of the stuff from a wrecked wooden ship floated. So much that the UK developed an elaborate system for establishing the respective rights of ship owners and scavengers-rescuers. This Wikipedia article gives a decent overview. Follow the link to Law of the Sea for more info.