There is a hairy hard shell (looking like a small kiwi), which naturally breaks, and under it there is brain-shaped red sweet flesh, half-covering the smooth black seed.
Picture taken in the Mediterranean coast of Israel in August
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$\begingroup$ Kinf of Morus : en.wikipedia.org/wiki/Morus_(plant) $\endgroup$– PhiberCommented Sep 7, 2018 at 16:33
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$\begingroup$ I think this tree is not native to the Mediterranean (cause I've never seen it). Was it located in a garden? Can you ask the owner of the house? $\endgroup$– RHACommented Sep 7, 2018 at 21:35
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$\begingroup$ @RHA I have seen it mostly in public places, along roads, in a hopital, etc. $\endgroup$– AnixxCommented Sep 8, 2018 at 6:28
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2 Answers
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Might it be Alectryon tomentosus?
To me, the fruit looks the same and also the leaves have the same look. 
It is an Australian tree whose common name is "wooly rambutan" and is used as a street tree as you observed. I did not find instances of this tree in Israel, though.
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Kind of Morus, I have seen this in Morocco and it is delicious : https://en.wikipedia.org/wiki/Morus_(plant)
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4$\begingroup$ No. Morus has many small seeds, this one has one big seed. Morus has no shell around the fruit. $\endgroup$– AnixxCommented Sep 7, 2018 at 17:47
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$\begingroup$ @Anixx I didn't say it is Morus, I said kind of Morus, which means Morus Family $\endgroup$– PhiberCommented Nov 15, 2018 at 21:35