Peaches was clearly distressed and trying to come down, but she kept trying to come down head-first which, of course, does not work. They tried everything they could think of to help Peaches down. With difficulty, they dragged a ladder through the woods and propped it up against the tree, but they could not get close enough to Peaches. They called the fire department which suggested that they spray her with water from a hose. Bad advice. That just caused Peaches to climb higher. They worked with Peaches till 2:00 AM before reluctantly giving up and going to bed. Sleep, however, does not come easily when you're distressed and can hear your beloved cat crying for you.

I set up my rope with little trouble, and, while I was doing so, Peaches actually climbed down her steep branch about five feet to the next fork. From there, she climbed down to the next branch at the red circle in the picture. At that point, the stem was almost vertical, and Peaches could go not down any farther. I began to climb up to Peaches. From the way she was acting, I was expecting her to be friendly, and I was very relieved to see that she actually was happy to see me. She sniffed my hand and readily let me pet her. She was perfectly comfortable with me, so this rescue should be easy unless I make a big blunder.
After we made friends and exchanged pleasantries, I gave her a chance to go into the carrier. She looked but did not go inside. So I opened a can of food and let her have a bite. She was mildly interested, so I put the food in the back of the carrier and gave her another chance to go inside. She stepped her front feet inside, but could not quite bring herself around to pulling those back feet in too. I gave her several chances to do it, but it became clear that she was not going to do it.
I put the carrier away and began preparing the bag. While I was doing so, she stepped onto my lap. We visited there a minute while I contemplated bringing her down that way. She was relaxed and the ride down would not be a problem, but I prefer to use two hands to control my descent, and I wanted Peaches contained at least until we got out of the woods. So, with her resting comfortably in my lap, I gently grabbed her by the scruff and lifted her while I pulled the bag around her. She took it all in stride without complaint.
I came down and handed Peaches over to Ginger. I followed them out of the woods to the cat house and released her inside there. Peaches was a little agitated at that point and needed some time to understand where she was and calm down. After a minute, she settled down and rested while Amber petted her. After a few minutes of that, Peaches got up, came over toward me and continued walking a complete circle around me while brushing against me. I interpreted that as a "thank you."
You're welcome, Peaches. It was my pleasure.