It's not based on lunar months as such, but the lunar cycle itself, particularly the full moon. The association of the Celtic tree or animal comes into play from the time of the full moon in that particular cycle, until the next full moon, so that part of the ogham interpretation as timing is valid, its just not known if it was ever used as a tool for divination.
Ok. So, I was born a little after Full Moon on 3rd September 1955 which would place me at the start of that Moon cycle.
If I was born 3rd September 1956 it would be just before New Moon
If I was born 3rd September 1954 it would be just before First Quarter
When does the Druid's year begin? March equinox which is a solar date? When does the Druid's cycle of 13 Lunar 'months' begin?
According to
http://www.joellessacredgrove.com/Celtic/tree.html
Today the counting of the Celtic tree calendar begins with the full moon nearest Yule. Perhaps it once began with the full moon nearest Samhain since, until the influence of the Norse, Samhain marked the beginning of the Celtic New Year. When you have pinpointed the full moon closest to Yule, count off the thirteen moons of the lunar year and mark them with their Tree Calendar names.*
Yule being Winter Solstice (Northern hemisphere) - a solar date
Yule: 22 Dec 1953, closest Full Moon: 20 Dec 1953, 3 Sep 1954 occurs in the 9th moon cycle - Hazel, which finished (in that year) with the next Full Moon 12 Sep 1954
Yule: 22 Dec 1954, closest Full Moon: 10 Dec 1954, 3 Sep 1955 occurs in the 10th Moon cycle - Vine, which started (in that year) with the Full Moon of 2 Sep 1955
Yule: 22 Dec 1955, closest Full Moon: 29 Dec 1955, 3 Sep 1956 occurs in the 9th moon cycle - Hazel, which finished (in that year) with the next Full Moon 20 Sep 1955
Using the same 'modern' calendar date, my birth date ended up in the Vine moon cycle but people born in years either side of mine were in the Hazel cycle.
PS: the same website says the Celtic New Year begins at sunset on October 31 - Samhain as noted above.
