Might mean (unverified): Measure of deviation from tree.
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We found one dictionary that defines the word treeness:

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  1. treeness: Wiktionary

Definitions from Wiktionary (treeness)

noun:  (nonce word) The essence of what it means to be a tree; the qualities that make a tree what it is.
noun:  (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.
noun:  (graph theory) The condition of being a tree; acyclicity and connectedness.

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