Half-sparse median-order embedding ($|f(T)\cap I|<|I|/2+1$)

A graph embedding into an ordered tournament is half-sparse when the displayed inequality holds for every proper final interval $I$. Reserving two extra tournament vertices allows extension at any outward-oriented leaf: the median order of a tournament supplies at least half of the later vertices as outneighbours of the parent's image, while half-sparsity leaves at least one of them unused. Adding one image while extending every old final interval by two preserves the inequality.