In my vocabulary, that's the root not the tip (I mention this since nodes regularly use the tip, but seldom the root). Nomenclature aside, you are truistically right in that one would have to download the full tree in order to locally verify it. And that's important; don't trust, verify.

For one-time verification purposes, there's no technical need to hold the full tree locally at the same time; each branch can be verified separately.

Also, once the validity of the full tree is established at a local node, little is gained from full revalidation. This is the principle which makes pruning possible in the first place.

Of course, the full tree still needs to exist, preferably at as many locations as possible. But not necessarly in an expensive (i.e. fast, locally complete and randomly accessible) form.