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Recent interest in treespaces as well-founded mathematical domains forphylogenetic inference and statistical analysis for populations of anatomicaltrees has motivated research into efficient and rigorous methods foroptimization problems on treespaces. A central problem in this area iscomputing an average of phylogenetic trees, which is equivalently characterizedas the minimizer of the Fr\'echet function. The Fr\'echet mean can be used forstatistical inference and exploratory data analysis: for example it can beleveraged as a test statistic to compare groups via permutation tests, or tofind trends in data over time via kernel smoothing. By analyzing thedifferential properties of the Fr\'echet function along geodesics in treespacewe obtained a theorem describing a decomposition of the derivative along ageodesic. This decomposition theorem is used to formulate optimality conditionswhich are used as a logical basis for an algorithm to verify relativeoptimality at points where the Fr\'echet function gradient does not exist.

Relative Optimality Conditions and Algorithms for Treespace Fr\'{e}chet Means