PHILADELPHIA—Teddy is on a diet, but for two Saturdays this month he’s delighted to eat his fill. The 200-pound Nubian breed goat needs to be at 180, his handlers say, to prevent arthritis pain in his knees. Yet, he recently munched contentedly on a buffet of pine needles and twigs at what has become an annual postholiday event here—the recycling of hundreds of discarded Yuletide trees into goat feed and, later, mulch.

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