I'd like to kill off the ivy on a very large, old ash tree, in the hope
of preventing it becoming overly festooned with the damned stuff [like
so many mature trees, around here].

So, I've cut the ivy climbers up the trunk (and left a large gap between
two cuts on each climber), and will keep an eye out to prevent new ones
growing.

My question: is there any truth in the old saw that ivy, once securely
established on a tree, will become genuinely parasitic, and will draw
sustenance from the tree itself?

Cheers
John