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What is treeness?

What is treeness? Here are some definitions.

Noun
  1. (nonce word) The essence of what it means to be a tree; the qualities that make a tree what it is.
  2. (genetics) The suitability of an evolutionary tree for representing the structure of a population; the degree to which a population structure can be accurately described as a tree of descent, with different branches evolving independently after they split.
  3. (graph theory) The condition of being a tree; acyclicity and connectedness.
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