Possible configurations of a tree having incomplete information

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I'm looking for a math application that would detect all possible trees that are compatible with an incomplete description. For being more concrete, I'm trying to detect all possible skeletons that are compatible with a known set of joints. At this moment I'm reviewing some papers on combinatorics group theory, but they're pretty abstract and it's hard to find applied use cases. Any hint in the right direction will be welcomed. Thanks in advance.