tree vs chase after

tree is a noun but chase after is not a noun.

As verbs, chase after is a hypernym of tree; that is, chase after is a word with a broader meaning than tree:

Other hypernyms of tree include dog, give chase, go after, chase, tag, tail, track, trail.

tree (noun)	chase after (noun)
a tall perennial woody plant having a main trunk and branches forming a distinct elevated crown; includes both gymnosperms and angiosperms
a figure that branches from a single root

tree (verb)	chase after (verb)
force a person or an animal into a position from which he cannot escape	pursue someone sexually or romantically
plant with trees	go after with the intent to catch
chase an animal up a tree
stretch (a shoe) on a shoetree