Triple rooted births are births caused due to Kamma done with all 3 wholesome roots: non-greed, non-hate, wisdom.
There are 3 types of births in the human world: Triple rooted, double rooted and rootless.
When you are doing a good deed, there are 3 stages: Purwa - preparation stage, Munchana - doing of the deed and the Apara - immediate aftermath.
During all these three stages, if your thoughts regarding the deed were accompanied with wisdom, the middle stage becomes a Triple rooted excellent(Ukkattha) Kamma. Such Kamma is able to produce triple rooted births.
During either the first stage or the aftermath, if your mind was lacking wisdom, the middle stage becomes a Triple rooted inferior(Omaka) Kamma. Such Kamma can only produce double rooted births.
During all three stages, if the mind was only accompanied by non-greed and non-hate but not wisdom, the middle stage becomes double rooted excellent Kamma. Such Kamma is able to give double rooted births. But if the first stage or the aftermath lacked either non-greed or non-hate, the middle stage becomes a double rooted inferior Kamma. Such Kamma can only produce rootless births.
It is said that to understand the Dhamma deeply, to attain the Jhaanas and the stages of enlightenment, one needs to be born with a triple rooted birth.
Individuals with triple rooted births have a further classification. Even though the good deed was done with all 3 wholesome roots, if non-greed was weak, that person is born with a triple rooted birth but also with a greedy personality. If non-hate was weak, that person is born with a triple rooted birth but also with a hot tempered personality.
If you get a double rooted birth, your mind won't be sharp enough to understand the deep Dhamma and significant meditative attainments would be beyond your scope within that lifetime. Nonetheless, your efforts to do good deeds, meditate and understand the Dhamma can result in you getting a triple rooted birth next life.
Rootless births are the most unfortunate type of births. If you get a rootless birth, on top of being dull minded, you can be blind, deaf, mute, limbless, deformed, transgender, autistic etc.
There is no concrete way to know if you have a triple rooted birth. Some teachers say that if you are able to concentrate on one thing without being distracted for at least 2 minutes, you most likely have a triple rooted birth. In any case, the best approach would be to stop wasting time wondering if you are triple rooted or double rooted and focus on cultivating goodness.
Further reading: https://www.wisdomlib.org/buddhism/book/abhidhamma-in-daily-life-ashin/d/doc3181.html