Whilst this is a long shot it immediately1 made me think of Rebecca's World by Terry Nation. It was first published in 1975 so might be a bit new and is 117 pages which also might be a few too many. However, it matches there only being one tree left in the world and there is someone "looking after" it, though they appear to be more like guarding it.
The excerpt from the below review is one of the better ones for describing the story:
It turns out that a very nasty man, Mister Glister, is the richest man in this world, and as he explains to our heroine most seriously, 'I've made it my life's work to make everyone poor', and he has done it by burning all the trees down in the world to feed his industry and make himself rich. Some of the shelters from the said GHOSTS are made of the wood from these trees, but given Mister Glister's over the top approach to construction, there stands only one tree in this world, and it is guarded by the GHOSTS.
The Write Thought, Book Review: Rebecca's World by Terry Nation
1. Although that is probably just because it was still in my head from when I read this answer to another story id question not too long ago.