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OK, wow... Gen 8:11 When the dove returned to him in the evening, there in its beak was a freshly plucked olive leaf! Then Noah knew that the water had receded from the earth. &
Neh 8:15 and that they should proclaim this word and spread it throughout their towns and in Jerusalem: “Go out into the hill country and bring back branches from olive and wild olive trees, and from myrtles, palms and shade trees, to make temporary shelters”—as it is written.
& "he sent out a dove" = "Go out into the hill country/mount"

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Since this is about the eighth day:
[the eighth day] This eighth day was not originally part of the feast, but an extra day commanded by the Priestly Law to be observed as ‘an holy convocation’ (Leviticus 23:36; Leviticus 23:39). Its celebration closed, as it were, the festival calendar of the Jewish sacred year. We do not hear of its observance in early times. As we might expect, it is not mentioned in the brief festival notice of Exodus 23:16. In Deuteronomy 16:13-17 it is not spoken of, it is only said ‘Seven days shalt thou keep a feast.’ In 1 Kings 8:65-66, we are told that after the Feast of Tabernacles Solomon sent the people away on the 8th day. In the Priestly Law, however, the observance of this 8th day is insisted upon as ‘a holy convocation,’ ‘a solemn assembly,’ on which ‘no servile work’ is to be done, ‘the eighth day shall be a solemn rest’ (Leviticus 23:36; Leviticus 23:39). It is interesting, therefore, to take notice that in 2 Chronicles 7:8-9 the observance of this 8th day is recorded, although not mentioned in the parallel passage, 1 Kings 8:65-66. The Chronicler recounts the celebration of the Feast of Tabernacles in accordance with his knowledge of the Priestly Law. Our passage agrees with the later observance and with the Priestly Law. The complete disappearance of the originally distinct character of ‘the eighth day’ is shown in 2Ma 10:6 ‘eight days … as in the feast of tabernacles.’
https://biblehub.com/commentaries/nehemiah/8-18.htm

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Olive Branch: peace, oil of anointing, the Holy Spirit, Mt of Olives
“if you were cut from what is by nature a wild olive tree, and grafted, contrary to nature, into a cultivated olive tree, how much more will these, the natural branches, be grafted back into their own olive tree” (Rom 11:24) and Gentiles, being “a wild olive shoot, were grafted in among the others and now share in the nourishing root of the olive tree” (Rom 11:17)