It's well know that Ezreal doesn't work like the usual marksmen. He's a caster and the common marksmen masteries aren't that good on him. That's why the 1st tree is out of the way, as well as the 3rd which is only for tanks.
The best choice is to take Windspeaker's Blessing, with Heal + Barrier as summoner spells (which benefit from your mastery, maximizing the synergy). This goes well with the fact that Flash is completly useless on Ezreal since you max E first for free escape. Think about Twisted fate, no one would take TP on him because your ult main use is to get back to lane.
Usually Ezreal is considered weak early, but you don't need to be a good sheep and buy this boring tear like everyone. An actual great first buy is Righteous Glory : it gives plenty of mana & mana regen, it makes you tanky (Ezreal is a safe ADC, so don't build only damage), and you gain mana when people hit you while gaining health when hitting champions.
Then, you can go for a Thornmail, it's the recommended item for Ashe as well which is another safe ADC, the armor without health synergize perfectly with the pure health from Righteous Glory.
Then, if you didn't already end the game, you can go for either Banner of Command or Redemption, depending on the game. Banner is great if the ennemy got poke, Ezreal model being pretty small, you can hide behind the buffed creep quite easily. Redemption is good combined with your ult : you can easily secure a baron/drake or one-shot people in the back line by launching your ult and using Redemption active on the same spot you're targetting.
You don't need more items to crush your opponent, but if the game really becomes long, you can use your remaining slots for boots, more speed is always good to come back from the base after a death.
So that's it, you may not see often that kind of Ezreal, because people fail to think outside the box. Your team may even try to report you, especially if they don't understand your strategy and fail to follow your steps. When that happens, link them this page and they will probably understand and apologize. Don't spread it too much tho, or else Ezreal will become 100% ban and nobody will be able to get free elo with this guide.