According to this question, when we have two 이, we can delete one and have one 이.
Is there special rules in these cases in writing and prounancing?
Is there any rules for other vowels?
Example: 이상한 아이이네요!
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Generally you can drop the 이 in 이다 when it follows a vowel. It sounds more natural to do it most pf the time. In formal writing, however, people tend not to drop 이. Also, we don't drop 이 when it is conjuagted with -ㅁ(nounifier). Also dropping 이 in 이에요 is considered grammatically wrong. Instead shorten it to 예요.
이건 나무다. This is a tree.
그는 이게 나무란 걸 알고 있다. He knows that this is a tree. (< 나무이라는)
이건 나문데. (< 나무인데)
누굴까? Who is it? (< 누구일까)
여기는 유토피아야. This is utopia.