Answer:
There is a bird nest in that tree.
Step-by-step explanation:
"They're is a bird nest in that tree." is incorrect because that would not make any sense at all since "they're" is a conjunction of the words, "they are", therefore "they're" can be eliminated.
"Their is a bird nest in that tree." is also incorrect because "their" is used to say that something belongs to someone, e.g. "It was their pencil bag." which states that the pencil bag belongs to them, whoever they are.
With the process of elimination and grammatic sense, we can conclude that the answer to fill in this sentence would be "there" for the sentence, "There is a bird nest in that tree."