I am sure that this subject has been covered. We have a lot of cat people here.
My cat went up a tree. A five year old male, and sterile. And couldn’t find an exit. I wasn’t worried. I thought he would eventually find his way down.
After 24 hours, my neighbor was worried. And he fetched a ladder from the neighbor. And brought the cat down.
If he didn’t do this, what would have been the likely outcome?
The cat would either manage to get down safely on its own, or not.
Thank you QtM. That link said a mouthful.
I prefer to let a higher source handle the outcome. Maybe it is population control?
It is difficult to see the big picture.
My cat went up a tree. I was worried about her, but didn’t have a suitable ladder. We were advised to leave a can of tuna on the ground to coax her down. I called a tree trimming company, since they have equipment to get up into trees, but they said they were too busy with their regular work. (And I think they were worried a cat might scratch one on them.) She did eventually come down, but the stats on cats climbing down after a day aren’t great. So, your cat might have come down, or might not have.
I once had a stray cat come up. He was very friendly. I assumed he was a drop off. I usually feed and worm and get shots for all drop offs. I couldn’t get my hands on him for more than a minute. He ate readily and occasionally would sidle up to my leg and take a pet or ear scratch. That cat would climb a tree and howl for hours. I took to rescuing him. I would get halfway down the ladder and he would bail. I kept thinking he would learn to either quit climbing or how to get down. Never did. He disappeared one day. I think he climbed a tree where he couldn’t be heard.
I have gone out of my way to help animals in distress. But I donate a lot more money to Planned Parenthood than to the ASPCA.
My cat, Graymalkin, has spent four nights in a row three different times up a tree. He finally learned how to back down. This was several years ago and there has been no problem since. He was pretty thirsty, but that was about it.
My ex-husband used to calm me down when our cats did this by telling me he’d never seen a cat skeleton in a tree. 
I frequently called tree services to get my cats down, and they would come if they were nearby or if they had the time. I’d tip them whatever money I had available.
Dennis was stuck in a dead tree with no limbs one morning, howling piteously. When I brought out food for the other cats, he came down for breakfast.
How did Dennis climb the tree if he had no limbs? 
True. Once they die, or just weaken enough, they generally fall out of the tree.
It quit working when I saw a raccoon carcass in a tree. Eventually vultures carried it off.
I believe he reppelled down a rope using his teeth. I was getting the cat’s breakfast at the time.
Man, you got to keep an eye on that Dennis.