There's an old joke that goes like this
Library User: 'Have you got a bookmark?'
Librarian: 'Yes thousands, and the name is John'.
Library is our first plant-based word. It came into Middle English as librairie and derives for Latin liber, 'book'. The Romans explained the name as being from the liber tree, whose bark had once been used for writing on.
Tree or not, liber is probably from PIE *lubh-ro- 'peel, leaf', which also shares a relationship with lodge, lobby and loggia, in the sense of an arbour or shelter with a roof of leaves or bark.
In modern Romance languages library has come to mean a bookshop or seller, and English library translates as e.g. French bibliothèque, from ancient Greek βιβλιος (biblios) 'book', after the Phoenician city of Byblos[1], which exported Egyptian papyrus to Greece. Or it could be the word was borrowed from Egyptian into Greek and the city got its Greek name from there. Or maybe the Greeks just garbled the city's old Phoenician name Gebal. Either way, from βιβλιος we get Bible, our second plant-based word.
In Old English a library was called a bochord, 'book hoard' or a bochus, 'bookhouse', both of which were eventually displaced in Middle English.
And that brings us to book, our third (probably) plant-based word. This has cognates in many Germanic languages; Gothic 𐌱𐍉𐌺𐌰 (boka)[2], Old Norse bok, modern German, Dutch: Buch, boek. The Germanic root these share is usually linked to the beech tree, because? runes where written on beech board or perhaps because book covers were made of beechwood. Wiktionary gives a possible PIE root which would also be the origin of Latin fagus 'beech' and tree-related words in a variety of IE languages.
But the connection with beech has often been disputed and is far from being universally accepted. The debate has swayed this way and that, and the pendulum has currently swung towards the beech tree explanation. You can read a good blog post about the debate here.
[1] Byblos is probably where the Western alphabet was invented.
[2] This is written in the unique Gothic alphabet, which has Unicode support. If you squint a bit you can see it says 'BOKA' or 'BUKA'.