 

In the early nineteenth century the notion of what constitutes proper academic credentials was in a formative stage. The record becomes spotty and even contradictory. Because it seems to be more direct, let's follow Plucker's line first. Plucker was the only student of his advisor.
Poisson and Fourier were academic siblings whose lineage traces back just one more generation:
It seems Lagrange only had three students: Fourier, Poisson, and Giovanni Plana (joint with Fourier). Plana seems to have had no students. Fourier only had two students: Plana and Dirichlet (joint with Poisson). Navier certainly studied with Fourier, but may not have formally been his student. So the entire Lagrangian line seems to pass through Poisson. Poisson had only three students. However these students and their almost 50,000 descendents played major roles in the development of mathematics in continental Europe and the United States.
The first of Poisson's students was Michel Chasles (1814), who advised only Hubert Anson Newton (1850 at Yale) and Gaston Darboux (1866 at ENS). Newton had only three students, all at Yale: Arthur Wright (1861), Charles Little (1885), and Eliakim Hastings Moore (1885). Wright and Little had no students, but E.H. Moore became the first Mathematics Department chairman at the University of Chicago where he had 31 students. One of Moore's students was Oswald Veblen (1903), who had 16 students, one being Alonzo Church, who had 33 students. Jointly with Veblen, E.H. Moore also advised R.L. (Robert Lee) Moore (1905), who had 50 students. Another student of E.H. Moore was G.D. (George David) Birkhoff (1907), who had 46 students, plus one wellknown son (Garrett Birkhoff).
Darboux had only five students: Emile Picard, Thomas Stieltjes, Emile Borel, Elie Cartan, and Stanislaw Zaremba (who was coadvised with Picard). The line of Emile Picard leads to Paul Painleve, Jacques Hadamard, Andre Weil, Maurice Frechet, Paul Levy, Mihailo Petrovic, Michel Loeve, Szolem Mandelbrojt, and others. Mandelbrojt and Loeve came to the United States. Thomas Stieltjes had no students. The line of Emile Borel leads to Henri Lebesgue, Henri Cartan, Jean Dieudonne, Laurent Schwartz, JacquesLouis Lions, Alexander Grothendieck, and other great names in French mathematics. The line of Elie Cartan leads to Vaughan Jones, now at Berkeley. The line of Stanislaw Zaremba leads to Waclaw Sierpinski, Antoni Zygmund, Jerzy Neyman, Karol Borsak, Samuel Eilenberg, and other names associated with Polish mathematics. Of course, Neyman, Zygmund and Eilenberg all end up in the United States.
The second of Poisson's students was Dirichlet (1827), who had only three habilitation students: Eisenstein (1845), Kronecker (1845), and Lipschitz. Eisenstein seems to have had no students, while Kronecker had six students. Lipschitz only had one: Felix Klein, who had 58 students . One was Fredinand Lindemann, who had 47 students. Most notable among them include David Hilbert, Hermann Minkowski, and Arnold Sommerfeld. David Hilbert had 75 students ! The most notable among them include Richard Courant, Hermann Weyl, Hugo Steinhaus, Erhard Schmidt, and Erich Hecke. The lines of Courant (through most of his 32 students ), Weyl (through Saunders MacLane), and Steinhaus (through most of the descendants of his ten students ) lead to the United States.
The last of Poisson's students was Liouville (1835). His only student was Catalan, who only had three students. One of them was Hermite, who had seven students. Among these was Pade, Poincare, Petrovic (joint with Picard), Stieltjes (joint with Darboux), and Tannery. It seems that Poincare had only two students, neither of which had students. Among Tannery's four students is Hadamard (joint with Picard). The Liouville line therefore largely reconnects with the line of Darboux.
The Lagrangian line seems for the most part to pass through Felix Klein and E.H. Moore. (Kronecker and Darboux are the only other routes.) The Kleinian and Moorian (?) lines were extremely prolific. It seems that through these routes, many Americantrained mathematicians are Lagrangians!
Acknowledgement. I had fun putting this page together with the help of websites of the Mathematics Genealogy Project (North Dakota State University) at "http://www.genealogy.ams.org" and the MacTutor History of Mathematics Archive (University of Saint Andrews, Scotland) at "http://wwwgroups.dcs.stand.ac.uk/~history". I welcome any comments or corrections.