In 1984, what does the Chestnut Tree symbolize? Why did Orwell choose that specific tree as a name for the cafe?

The chestnut tree is often understood as a symbol of justice and honesty. We also hear the phrase "an old chestnut" to describe a story that has been repeated many times.

More specifically, Orwell connects The Chestnut Tree cafe to a song Winston hears twice on the telescreen during the course of the novel:

Under the spreading chestnut tree/I sold you and you sold me/There lie they and here lie we/Under the spreading chestnut tree.

Winston hears it the first time when he happens into The Chestnut Tree cafe before his arrest, when:

He [Syme] said things that would have been better unsaid, he had read too many books, he frequented the Chestnut Tree Cafe, haunt of painters and musicians. There was no law, not even an unwritten law, against frequenting the Chestnut Tree Cafe, yet the place was somehow ill-omened.

Winston hears the song again when The Chestnut Tree has become his home away from home after his arrest and release from prison, but this time only the first two lines: "under the spreading chestnut tree/I sold you and you sold me." These lyrics bring tears to his eyes.

Clearly, the song and the cafe represent Winston's betrayal of Julia and hers of him, and through it, everything they loved: "I sold you and you sold me."

Further, Orwell uses the line "They lie there and here we lie" to pun on the meaning of lie: in the song, it means lying down, but in the world of Oceania it means telling lies. 

If Orwell is thinking of "old chestnut" as a story repeated many times, then we have to believe that what Julia and Winston have undergone is an old story: they are not the first nor the last to be broken by the regime. The name also evokes the "old England" Winston longs for, but ironically the cafe represents the new world of the regime.

My sense of the novel is that Orwell was specifically referring to the song in naming the cafe, and not looking to the deeper symbolism of the tree as representing truth and justice, but certainly that deeper symbolism lurks ironically in the background, for in Oceania, the chestnut tree, once the symbol of truth and justice, is twisted into the symbol of lies and injustice.