Genesis 7:13-16
In the same day Noah, and Shem, Ham, and Japheth, the sons of Noah, and Noah’s wife, and the three wives of his sons with them, entered into the ship; they, and every animal after its kind, all the livestock after their kind, every creeping thing that creeps on the earth after its kind, and every bird after its kind, every bird of every sort. They went to Noah into the ship, by pairs of all flesh with the breath of life in them. Those who went in, went in male and female of all flesh, as God commanded him; and Yahweh shut him in.
Here we read that the birds were on the Ark as well as every creeping animal. There was a lot of DNA represented there. Some people doubt that every kind of animal could have fit on the Ark, but remember that the Ark had three levels and each level was bigger than a football field. God is able to bring animals that were of the age that they would fit together. The dinosaurs were also there at the time but they were probably small ones. They may have been babies or young compared to the huge ones we see as fossils.
Not every breed of animal needed to be represented either. We know that differences can happen any time later to create vastly different appearances for things like dogs and cats. If God brought the animals with the best representation of complete DNA, then the breeding would bring differences back again later.
It is interesting that God shut Noah in. Noah would never be able to blame himself for closing the door on the rest of mankind. It was God’s decision and time was up. God has promised us protection if we are in Jesus, but time will be up someday and it will be too late after that point.