In fact, though, there is a disagreement as to which day the new year of the trees falls on: the 15th of Shvat, like we say, or the first of Shvat. This disagreement is brought in the Mishna, Rosh Hashana 1:1:
There are four new years.
The first of Nisan is the new year for kings and holidays.
The first of Elul is the new year for animal tithes. R' Elazar and R' Shimon say: The first of Tishrei.
The first of Tishrei is the new year for years, sabbaticals, and jubilee years, for planting and vegetables.
The first of Shvat is the new year for the tree, says Beit Shammai. Beit Hillel says: The 15th [of Shvat].
What is the basis of the disagreement between Beit Shammai and Beit Hillel about the date of the final new year?
Beit Shammai's reasoning appears to be simplest. The other three new years in the Mishna all fall on the first of the month. Logically, it seems months should be subsets of years, so that a month should not be split between different years, and years should begin on the first of the month. We know that's the case for three out of the Mishna's four new years, and it makes sense to say the same for the fourth new year.
Beit Shammai, therefore, seems to have a strong argument for their date. Why does Beit Hillel reject this argument and choose a different date?
It seems to me that Beit Hillel sees the new year for trees as primarily being in the category of holidays, not new years. The 15th of the month brings up associations of the holidays Pesach and Sukkot, which also begin on that date. Perhaps Tu Bishvat should be seen as a holiday as well.
(Why are holidays so often on the 15th? Here's my guess. Holidays are times of communal events and interaction. On the 15th, when there is a full moon, people can get together at night as well as during the day.)
In addition to their dates, Pesach and Sukkot are similar in that they each last 7 days and are dates of aliyah leregel. Furthermore, they take place in the 1st and 7th months of the year. This places them exactly 6 months – half a year – apart from one another.
Interestingly enough, there is also a holiday exactly 6 months away from Tu Bishvat: Tu BeAv. Even more interestingly, its basic theme may be about trees, just like Tu Bishvat. The Gemara (Taanit 30b-31a) brings several explanations for why Tu Beav was celebrated. The explanation about the tribe of Binyamin is best known, but the final explanation given is that on Tu Beav they would stop cutting down trees to be used as firewood on the altar. After Tu Beav, the dry heat of summer was not sufficient to prevent worms from infesting the wood and making it invalid for Temple usage.
The holidays of Pesach and Sukkot are closely tied to the grain harvest. Around Pesach time the grain is growing to maturity, and the harvest does not begin until after Pesach (the law of hadash). Sukkot is called “the holiday of gathering” (Shemot 23:16) and corresponds to the gathering in of wheat stalks which had been left all summer to dry in the field. Thus Pesach marks the beginning of the agricultural year for grain, while Sukkot marks the end of that year.
It is possible that Tu Bishvat and Tu Beav have a similar function, for trees rather than grain. Tu Bishvat, as we know, marks the beginning of the new growth of trees. Tu Beav, according to the explanation we brought, marks the end of the period in which trees are cut down. Just like Pesach and Sukkot delimit the agricultural year for grain, Tu Bishvat and Tu Beav delimit it for trees.
In Devarim 8:8, the blessedness of the land of Israel is described in terms of the seven species which grow in it: “A land of wheat and barley, and grapes and figs and pomegranates; a land of olive oil and [date] honey”. The first two species are grains, while the remaining five are trees. Both grains and trees are responsible for the agricultural bounty we enjoy when living in the land. It seems that likewise, both grains and trees are linked to pairs of holidays, whose dates serve to define the produce's respective growing seasons.
Partly drawn from here, and some source I can't now locate which made the 15th->holiday connection. It's also my 500th post on this blog, mazal tov!