Hi Nichola, During Summer it's a pretty safe bet that any tree without any greenery or leaves on it is a dead one. In Winter it can be a little harder to tell the living from the dead at first glance, but a closer look will often reveal tiny buds or healthy looking green bark on the tips of a living tree. The ultimate test is to chip away a little of the bark on the tree. If the wood beneath the bark, or the inner bark itself is moist with sap, green and healthy looking then the tree is alive. Dead trees tend to have hard bark (if any) that is dry and hard all the way through to the good wood beneath. Always check ALL the way around a tree for signs of life as you will often come across trees that have an apparently dead side but are alive and kicking on the other side of the trunk, supporting a healthy canopy of leaves above. This can be caused by fungal infections, natural ageing, or even lightening strikes. As with so many aspects of bushcraft and woodskills, the best way to learn is to buddy up with someone who already "knows their oak from their ash" or to simply experiment all you can and learn by your own experience.
The fallen tree that Ray mears cut wood from was probably a willow. These often fall in the soft (wet) ground they like to grow in and take root again from the entire side of the trunk that's touching the ground. They throw up new shoots from all along the previously shoot-less trunk and form almost a natural hedge all on their own. Taking wood from sucha willow will do it very little, if any harm, as the tree will simply throw out new shoots to replace any you take away. If you want a piece of wood from a willow but it means cutting more than you need (perhaps you can't reach the bit you want without cutting it off lower down) take the limb off that contains the bit you want, then re-plant the excess by simply pushing the lower end of the sticks well down into the soft mud. Willow has amazing regenerative powers, and those little sticks will, 9 times out of 10, simply take root and grow into new trees as long as they are in damp enough soil.
I once replanted a whole stretch of riverbank this way after a temporary lorry road had been put through the area for gravel extraction. Once the road had been taken back up, I cut loads of willow "withies" (small sticks growing up from those fallen trunks) and stuck them in the soft marshy ground alongside the river. I am now able to walk amongst towering willows again where the gravel people would have left the riverbank bare and empty.