What grammar is involved in a sentence such as "Would you like to climb a tree?" [closed]

Yes, it's a verb. Like many verbs having to do with mind and emotion (think, believe, want, wonder, know, enjoy, etc.), like takes a direct object of a complement clause that explains the particular mental or emotional state the verb indicates.

Syntactically, English has four types of complement clause. Which kind of clause gets used is determined by the verb. Like can take either an infinitive or a gerund complement.

• I liked/wanted/needed/tried to read comic books. (infinitive complement clause)
• I liked/enjoyed/hated/tried reading comic books. (gerund complement clause)
• I think/believe/know/doubt/see that he did it. (tensed that-clause complement)
• I know/wonder/heard why he mentioned it. (tensed Wh-clause complement)

But like is marginal at best with tensed complements:

• ??I like that he did it.
• ?*I like why he mentioned it.

tl;dr - there are lots more like like; most sentences are complex, and object complements are common.

Yes, it's a verb. See https://www.merriam-webster.com/dictionary/like

like verb \ ˈlīk \ liked; liking Definition of like (Entry 1 of 9) transitive verb

1a : to feel attraction toward or take pleasure in : ENJOY likes baseball b : to feel toward : REGARD how would you like a change 2 : to wish to have : WANT would like a drink