Wings of Song by Lloyd Biggle Jr.
This seems to be an obscure story. Apart from its magazine publication it has only appeared in a few obscure anthologies none of which I have. I found this almost by accident in a PDF of The Magazine of Fantasy and Science Fiction November 1963 that I'm reluctant to link as I suspect it isn't a legal upload. However user14111 has managed to find a copy on archive.org (thanks :-).
Anyhow, the protagonist Karl Brandon visits a curiosity shop and finds a violin made from an unknown material called wood. However the violin is broken and no-one knows how to fix it. Eventually a reference is found in the bowels of a library:
“Information about wood, sir. I’m afraid not very much is known about it. But I did find one thing. About a hundred years ago, on the planet Beloman - that’s in the Partu Sector - there was a man who gave his occupation as woodcarver."
Brandon goes to the planet Beloman where the last tree in the galaxy is still standing:
It stood near the house, a straight, rough-textured finger pointing upwards into a top- heavy crown of green foliage.
“Is it . . .”
Parker nodded. "A tree.”
“I thought there wasn’t a tree left in the galaxy!”
“Evidently,” Parker said, “there's one left.”
The caretaker is Peterson, but he refuses to repair the violin and Brandon is forced to leave empty handed. Eventually the violin is repaired using wood from a wooden box found in a museum, and Brandon returns to Beloman to show Peterson the repaired violin. But:
"The tree’s gone,” Parker said.
"He said he was about ready to use it,” Brandon said.
They headed directly for the workshop, and Brandon had his hand on the door when a call stopped him. The young woman they had met on the first visit hurried towards them.
“What was it you wanted?” she asked.
“We’d like to see Mr. Peterson,” Brandon said.
“I’m sorry. Father is dead. He died a month ago.”
Brandon could manage no more than a deep breath.
“I’m sorry,” the woman said again.
“I’m sorry, too,” Brandon said.
They turned away. Slowly they walked back to the aircar. Slowly they flew away.