*** SPOILER *** Plot/logic hole.
Only people who are on Death's list can see him and Pud could see him before Death was stuck in the tree. So, Pud was on the list...BUT how would Pud have died if Death had not fooled him into climbing the fence? The fence did not exist before Death became stuck and Death was tricked into getting stuck in the tree.
Even if Pud was destined to fall off some other fence without Death's encouragement, the fall in itself is not enough to kill Pud, in the end he's given a mercy killing by Gramps(who, if things had gone according to plan, should have died much earlier from a heart attack, it seems).
Therefore: Death killed Pud out of boredom or spite and had planned to do so from the beginning.
Or: Pud was destined to die shortly after Gramps and Death just thought to speed things up.