Taanis 4:4 introduces the concept of Karbanos Eitzim. The Gemara (28a) explains what was so special about these days:
ת"ר למה הוצרכו לומר זמן עצי כהנים והעם אמרו כשעלו בני הגולה לא מצאו עצים בלשכה ועמדו אלו והתנדבו משלהם
The Sages taught: Why was it necessary to state the times for the wood offering of priests and the people? They said: When the people of the exile ascended [to Jerusalem in the beginning of the Second Temple period], they did not find enough wood in the Temple chamber [for the needs of the altar]. And these families arose and donated from their own [wood to the Temple].
וכך התנו נביאים שביניהן שאפי' לשכה מלאה עצים יהיו אלו מתנדבין משלהן שנאמר (נחמיה י, לה) והגורלות הפלנו על קרבן העצים הכהנים הלוים והעם להביא לבית אלהינו לבית אבותינו לעתים מזומנים שנה בשנה לבער על מזבח ה' אלהינו ככתוב בתורה:
And the prophets among them stipulated as follows, that even if the entire chamber were full of wood, these families would donate from their own property [on these specific days], as it is stated: “And we cast lots, the priests, the Levites and the people, for the wood offering, to bring it into the house of our God, according to our fathers’ houses, at appointed times year by year, to burn upon the altar of the Lord our God, as it is written in the Torah” (Nehemiah 10:35).
So these days were days on which people had donated wood previously. Now, when were these days? Taanis 4:5 tells us:
זְמַן עֲצֵי כֹהֲנִים וְהָעָם, תִּשְׁעָה. בְּאֶחָד בְּנִיסָן, בְּנֵי אָרַח בֶּן יְהוּדָה. בְּעֶשְׂרִים בְּתַמּוּז, בְּנֵי דָוִד בֶּן יְהוּדָה. בַּחֲמִשָּׁה בְאָב, בְּנֵי פַרְעֹשׁ בֶּן יְהוּדָה. בְּשִׁבְעָה בוֹ, בְּנֵי יוֹנָדָב בֶּן רֵכָב. בַּעֲשָׂרָה בוֹ, בְּנֵי סְנָאָה בֶן בִּנְיָמִין. בַּחֲמִשָּׁה עָשָׂר בּוֹ, בְּנֵי זַתּוּא בֶן יְהוּדָה, וְעִמָּהֶם כֹּהֲנִים וּלְוִיִּם וְכָל מִי שֶׁטָּעָה בְשִׁבְטוֹ, וּבְנֵי גוֹנְבֵי עֱלִי בְּנֵי קוֹצְעֵי קְצִיעוֹת. בְּעֶשְׂרִים בּוֹ, בְּנֵי פַחַת מוֹאָב בֶּן יְהוּדָה. בְּעֶשְׂרִים בֶּאֱלוּל, בְּנֵי עָדִין בֶּן יְהוּדָה. בְּאֶחָד בְּטֵבֵת שָׁבוּ בְנֵי פַרְעֹשׁ שְׁנִיָּה.
The times [of the delivery] of wood [for the altar] by priests and people, were on nine appointed days; on the 1st of Nissan, the family Arah ben Jehudah [delivered]; on the 20th of Tamuz, the family of David ben Jehudah; on the 5th of Ab, the family of Parhos ben Jehudah; on the 7th, the family of Jonadab ben Rechab; on the 10th, the family of Sinha ben Benjamin; on the 15th, the family of Zatoo ben Jehudah, and with them priests and Levites, and all those who did not know from what tribe [they were descended], also the family of Gonebé Eli, and that of Kosehai Kesignot; and on the 20th, the family Pachat Moab ben Jehudah; on the 20th of Elul, the family Adeen ben Jehudah; on the 1st of Tebet, the family Parhos, for the second time.
So there were nine days on which wood was donated. Five of them were in the month of Av, following a sixth on the 20th of Tammuz. The other ten months of the year have to make do with just three donations.
Why are these dates so bunched together? If people donated when they ran out, why did some families donate so little as to leave the Beis HaMikdash with as little as two days’ worth of wood in some cases? And if people donated when they so chose, not necessarily when the Beis HaMikdash ran out, then why did they decide to bring on these days and not on others?
I should highlight the donation of Adeen Ben Yehudah. They donated on the 20th of Elul, and the following donation wasn’t until Rosh Chodesh Teves. If they donated when they needed, then they not only donated nearly 100 days worth, but that period overlaps Rosh HaShanah, Yom Kippur, and Sukkos, plus several Roshei Chadashim, making this the period with the most Karbanos of any of these donations. Compare that to Parhos Ben Yehudah, who brought two days’ worth; no wonder they went back and donated a second time to cover Teves through Adar.