Bagh e Jinnah
It is the oldest part in town with very old oak and banyan trees. Walking trail is also good and has the best variety of unique flora. There's a beautiful Library right next to it, which has served as a Gymkhana a few decades ago.
The British were so homesick for their country gardens that they recreated one right in the heart of Lahore. Lawrence Gardens has had a patriotic rebrand, and remains a lush and lovely oasis to linger with a good book.