I like this series of pictures (from the Cotley Hunt hedgelaying competition) because it gives a clear idea of how pleaching (or plashing) works.
The first step is to use the axe and then the billhook, to chop part of the way through the base of the trunk. The trick here is to cut precisely, just far enough into the tree to allow it to be laid horizontally, but to leave the bark, some cambium (the layer immediately beneath the bark) and a little sapwood (the living cells that surround the heartwood) so that the horizontal part of the tree will survive the process.
Then the tree is gently pushed down to the hedge:
The tree remains attached at the base and thus is able to stay alive.
Now the tree can be bound into the hedge – this is a Devon style hedge so crooked stakes will be used, with no horizontal heathering.
This is a good view of how the base of the tree is cut: