Since I catch a lot of tree trout, have a couple of devices that helps retrieve the fly (and/or flies left by others). Small enough to carry in a pocket or vest and easy to use. Works if fly is still attached to the tippet or if the tippet has been broken. I'm sure there are others out there besides what I have pictured.
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I like this little device if the fly is hanging from a tippet so that you can catch the fly by the magnet and cutter. Doesn't work if the fly is wrapped up against the branch.
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Another device for tree trout (bass, sunfish, etc.) I don't much care for it as the little metal point that fits into the tiptop comes out really easy and it there is very much foliage then it gets knocked out too often. The advantage over the catch-a-lure is that it allows you to pull the branch down to where you can release your fly.
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I copied this off a similar commercial item, which I prefer, even if the fly is dangling. Repurposed the hook part from an old bungee cord, a piece of broken rod that has one end plugged, used super glue and heavy thread to join the rod piece to the hook and added a cord bout 3 feet longer than my longest fly rod. The contraption will stay on the rod until you have positioned it on the branch and removed the rod.
There is another device that has racheting blades that will cut the branch if it is not too big, but I have not tried it so unsure how it really works in use.