So the bully was just spouting nonsense by calling it "the sacred tree"?
I wouldn't put it past them, him being a bully and all that.
But what, then, is it that triggers the tree to know to bind someone to a pact/oath/deal?
Kind of. The Alzer Republic has been calling it that for untold generations, so he just doesn't know better. In short, they don't understand what they have, but that does not prevent them from using it. That makes sense in the context of the story, since it was previously established that all lost tech is by definition incomprehensible to the current humanity, though not necesarily unusable or particularily powerful. For example, conventional, combustion-based firearms have been found in various ruins in that world and no one (other than perhaps reincarnators) knows how they are able to function. They aren't in any real demand though, since the magi-tec versions they already had, that shoot magic bullets, are considered superior. Same thing with Leon's Arroganz - when the school first saw it during his duels they knew right away it was lost tech, because similar ones were discovered by adventurers. They still mocked him for it being too bulky and unwieldy to be a challenge for the speed-optimized magic versions of the prince's entourage. They didn't expect the performance to be anywhere near to what it really was.
The tree responds to the bearers of it's crest, in this case the nobles from the country's great houses, Pierre is one of them. This is why they have absolute authority and even powers beyond the magic available to all New Humans. In other words, a non-bearer has almost no chance against them in all-out, conventional combat - this is why no other country has ever beaten Alzer, when fighting within the tree's radious of influence, it allows them to annihilate entire armies.