Definition: Balanced tree datum

A tree datum consists of a finite tree on vertices $V=I\bigsqcup X$, with external vertices $X\ne \varnothing$, smooth vertex weights, and edge distributions whose wavefront covectors are balanced: an edge from $a$ to $b$ has endpoint covectors $(k,-k)$. Assume the closed graph support determined by all weights and edge supports is compact. Finite differentiations and species contractions are specified.