As an entrepreneurial venture, this could be profitable. Scipione Ammirato, the Italian writer and historian, set up a workshop in Florence and turned out a whole series of them in cooperation with artist-engravers (Congedo, 216). On spec, he sent a family tree of Henry III to the French royal court. He received a reward of 500 gold ducats from Paris for it (Congedo, 274).
The oldest of Ammirato's trees is probably that of the Hapsburgs of Austria, engraved in 1576. It shows the tree on a high hill over a bay (probably representing Trieste) where a great naval fleet rides at anchor. The original copper plate still exists in Florence, according to the Italian register of cultural heritage, though it is not stated who owns it. An original print from it was sold by the antiquarian bookseller Gonnelli a few years ago as part of a set for 300 euros:
The type continues with Ammirato's 1580 book Delle Famiglie Nobili Napoletane. which contains eight double-page and five single-page engraved illustrations of genealogical trees. Each contains some kind of landscape in the background that can be connected with the dynasty. Here is my plot of part of the tree of Marzani, who were big shots in the Spanish-ruled Kingdom of Naples, from that book:
The conscious idea of presenting a complete family line connected by a woody trunk first shows up in southern German woodcuts in the late 15th century. [Later note: Dr Volker Bauer (see below) has kindly pointed out a magnificent early example in BSB incunable 2 Inc.s.a. 1264 from about 1475 digitized here. There is also a most remarkable tree of heraldic arms in the 1492 Cronecken der Sassen, GW 4963.] This reaches its finest flower in the Ehrenpforte engraved in 1515 by Albrecht Dürer (there's a fine reconstruction of this on Virtuelles Kupferstichkabinett). But Dürer's has no branches.
The tree as a recognizable category of art, a product where artist and customer know what to expect, only shows up later in the sixteenth century. It looks semi-natural, has a bottom root and clearly tiered generations. The oldest example I can find is Robert Peril's 1535 tree of the Hapsburgs made at Antwerp (online: Boijmans Collection).
Examples later than Ammirato's include a fine 1586 tree of the Kings of Saxony by Lorenz Faust which is labelled "Stammbaum," perhaps the first documented use of that word in the German language (the link is to the MDZ in Munich). [Later: Note however a 1515 illustration title "Bawm vnnd Außlegung der Sypschafft ..." here.] The type's later development in Germany and embrace of tree shapes other than oaks has been researched by Volker Bauer of the Herzog August Bibliothek.
But I cannot find any trees of living families from the first third of the 16th century. Has anyone got suggestions?