Inspired by this answer:
As far as I can tell, there is no mention of Noach bringing plants into the ark. Moreover, it appears Noach had nothing to do with the survival of the plants, given that the dove brought back the olive tree leaf after the flood, before anyone had a chance to plant anything.
So, how did plants survive the Mabul?
The Classic Questions to Bereshit 8:11 in the Gutnik Chumash brings several different opinions on this matter (while specifically addressing where the olive branch came from).
Rabbi Levi says (Bereshit Rabbah 33:6) that the floodwaters did not fall in the Land of Israel. If so, even if all the plants in the rest of the world were destroyed, that would not mean the extinction of the plants.
The Ramban says that since the whole world was filled with water, there weren't strong currents that would uproot trees, and the trees survived the flood.
The Abarbanel says that the branches from the destroyed trees were floating in the water. As the flood subsided, the branches took root and started growing again.
Ralbag explains that Noah brought seeds with him on the ark to replant the world afterwards, and he provides three reasons why the Torah doesn't mention this:
וראוי שתדע שמיני הצמחים אשר יתכן הפסדם מפני המבול השגיח השם ג"כ בהם בזה האופן ששם נח מהזרעים ההם בתיבה בדרך שישאר מהם לזרע בארץ אחר המבול ולא זכרה זה התורה בביאור לפי שהרבה מהצמחים לא נצטרכו אל זה כמו האילנות ולזה השמיטה התורה זכרו עם שזה דרך התורה לזכור קצת הפרטים ולעזוב קצת והנה זכרה הב"ח להיותם יותר נכבדים מהצמחים והיותר נכון שנאמר שזה נכלל בתורה במה שאמרה והיה לך ולהם לאכלה ר"ל כי יחויב מזה שיספיקו הזרעים ההם להיות מהם מזונות הב"ח בתיבה ואחר צאתם ממנה וזה יהיה כשיזרעו בארץ כי אי זה תועלת יהיה כשיהיה להם מזון בתיבה ואחר שיצאו מן התיבה לא יהיה להם מזון וימותו זה המאמר מבואר הבטול כי הצלתם שם יהיה אם כן לבטלה ולא יתקיים מה שרצהו השם יתע' לחיות זרע על פני כל הארץ ולזה הוא מבואר שכבר יחוייב מזה שישאר מכל הזרעים לזרע בארץ אחר צאתם מהתיבה
And it is proper that you should know that the species of plants that could be destroyed by the Deluge were watched over by God as well in this manner, that Noah placed some of those seeds in the ark in a manner that there would remain from them to plant the land after the Deluge. And the Torah does not clearly mention this because many plants, such as trees, did not require this. And therefore the Torah left out mention of it. Additionally, the style of the Torah is to mention some details and leave out some. And it mentioned the living creatures because they are more honorable than plants. And what seems most correct is to say that it is included in the Torah when it said "it will be for you and for them to eat", meaning to say that it necessarily follows from this that these seeds should suffice to provide sustenance from them for the living creatures in the ark and after leaving it, and this would be by planting them in the ground. For of what benefit would it be for them to have sustenance in the ark but not have sustenance once they leave the ark, and they would die? This idea is clearly invalidated, for if so saving them would be for naught and it wouldn't fulfill what God may He be exalted wanted to cause seed to live on the entire face of the earth. And this is why it is clear that it already necessarily follows from this that there must have remained from all the seeds to plant in the ground after leaving the ark.
Does the Bible ever exaggerate?
Why didn’t Noah save the plants? Well, Noah didn’t save the fish either. Needless to say, the fish would have survived the flood even though the Bible states that all creatures were corrupted upon the face of the earth. One way to interpret it is that the Bible always exaggerates to make a point. It obviously meant that most people were sinners and that the flood was of natural consequences, which is to say that people didn’t pay attention to the signs of the possibility of a flood and remained where they were. Acts have consequences that can have long-lasting effects even for generations.
Was there a world wide flood?
So, what of the plants? Now that we know that not every plant was corrupted and in need of eradicating, we can safely assume that not all were drowned. Indeed, the Bible doesn’t state that all plants drowned for a dove brought back a branch for Noah, indicating that land was nearby and that it was time to land. It is possible that the flood was a local event. This would explain why people and animals inhabited the world over before and after the flood. To support this claim I will quote Rabbi Marc Shapiro who says that it would be impossible to image a certain animal only inhabited to its region would have lived elsewhere, especially if the animal could not sustain in the region, not to mention the lack of evidence.
Summary
Now that we know the flood was local and that there was no need to kill all living things since it is not possible to imagine that everyone with the sole exception of one family were guilty. The Torah itself makes this very claim by citing the doves’ return to the boat with a branch, indicating that it was a local flood.
