The Leatherstocking arc
Across five novels, Cooper traced the life of one character — Natty Bumppo, the white frontiersman raised by Mohicans — from old age (The Pioneers) back through his vigorous middle years (The Last of the Mohicans, The Pathfinder) to his death on the prairie (The Prairie) and then, in the last book published, back to his youth (The Deerslayer). The arc is not chronological in publication order, and that itself is part of the teaching: the same life seen from different windows.
The deeper relational current is the bond between Natty and Chingachgook, sachem of the Mohicans, and Chingachgook's son Uncas. Across the novels they hunt together, fight together, mourn together. Natty learns the forest as Chingachgook knows it; Chingachgook trusts Natty with his line. The chosen-brotherhood teaching is older than Karl May; it was Cooper's first.