The Collaborative International Dictionary of English v.0.48:

Burgeon \Bur"geon\, v. i.
   To bud. See Bourgeon.
   [1913 Webster]

WordNet (r) 3.0 (2006):

burgeon
    v 1: grow and flourish; "The burgeoning administration"; "The
         burgeoning population"

Moby Thesaurus II by Grady Ward, 1.0:

95 Moby Thesaurus words for "burgeon":
   acrospire, augment, bine, bloom, blossom, blow, bough, branch,
   branchedness, branchiness, brew, bud, build, burst forth, deadwood,
   develop, effloresce, enlarge, expand, flagellum, flourish, flower,
   flower bud, fork, frond, gather, gemma, gemmate, gemmula, gemmule,
   germinate, grow, grow rank, grow up, heighten, hypertrophy,
   increase, leaf, leaf bud, leaf out, leave, limb, luxuriate, mature,
   mount, multiply, mushroom, offshoot, outgrow, overdevelop,
   overgrow, overrun, overtop, plumule, procreate, pullulate,
   put forth, put forth leaves, put out buds, ramage, ramification,
   reproduce, riot, root, run up, runner, sarment, scion, shoot,
   shoot up, slip, snowball, spear, spray, sprig, spring up, sprit,
   sprout, sprout up, stolon, strike root, sucker, switch, take root,
   tendril, thallus, thrive, tower, twig, upshoot, upspear, upspring,
   upsprout, vegetate, wax