While the dragon family tree is just as in bred as the Targaryan line, it's not nearly as fleshed out and there's large gaps in canonical knowledge, I would take that family tree with a huge grain of salt.
We know there are male and female dragons and we know they rarely have clutches of eggs once they are of age to reproduce, more than once per lifetime, but it's very rare (not annually like other animals with mating seasons). This is just my conjecture here, but it stands to reason, they are very long lived creatures, ~200 year life span, with virtually no predators, so from an evolutionary sense (yeah its a fantasy world...) but it stands to reason they have clutches of eggs at most, every human generation (~40 years) or so. This part really isn't that fleshed out. For the most part each she-dragon lays a clutch of eggs and multiple dragons are hatched all at once, and if you look at the tree it appears that the dragons only have 1 clutch per lifetime. But we know that dreamfyre laid multiple clutches (a clutch of 3 of her eggs are stolen by the lesbian lover of her Rider and they ended up in the free cities). Syrax lays a 2nd clutch of eggs, one of those eggs Daemon tries to steal to give to Mysaria when she is pregnant with his child, but that egg is taken back and given to his daughter Rhaena after her first egg didn't hatch and later on it hatches and is named Morning, the last female dragon ever hatched, with no male dragons around to mate with her when she reaches puberty. The last dragon ever hatched (named the last dragon) probably came from that same clutch of eggs but it died within a day or so of hatching.
We know in the book the dragons are divided into generations, where a she-dragon has laid a clutch of eggs and X number hatched, that is considered a generation. When there's only 1 male and 1 female laying eggs (like Balerion+Vhagar) then it's easy to tell which dragons came from them. In that next generation however, when there were 2 mated pairs (and targaryans were actively collecting their eggs) it's harder to tell which dragon which egg came from, unless it's specified in the books.
notable errors with that reddit link, Vermithor was lifelong mate of Silverwing, just as their riders (King Jahaerys and Alysanne) were.
Quicksilver and dreamfyre were mated. Drogon, Viserion, and Rhaegal all came from Dreamfyre's line...actually, looking at that chart, they flipped it.
Vermithor + Silverwing produced the Orange lineage, minus the 3 dragon eggs from GoT.
Quicksilver and Dreamfyre produced the red lineage + the 3 dragon eggs from GoT.