why dont birds fall off trees when they sleep?

They have a tendon in the back of their leg that, when it tightens, holds the toes tightly curled. In most birds it tightens when the light is dim enough (dusk). It's called the roosting reflex.

Birds never enter deep sleep. If they begin to lean foward or backwards during sleep they adjust themselves

The bird's foot closes and grasps automatically as the ankle and knee joints are bent, This grasp cannot be released until the limb is straightened again.

My question is: Don't birds do a bit of sleepwalking on branches?

When a bird bends its knees the paws grip tightly.

They can't even let go of the branch without stretching their legs.

Their claws dig into the bark to keep them up.