First, the Utility Tree :

Then I'll explain my choice into the Defence Tree :

Into the offence tree, we will spend our points like so :

First, we start with some nice 8.55 physical damage from Greater Mark of Attack Damage because of two things : first we need a bit more of early game damage in trades, then Poppy damage doesn't come from Ad, it's hybrid damages, so ArmorPen won't help that much.

Then we pick up the Greater Seal of Armor, there is pretty much nothing to explain here.

The cooldown reduction increase Poppy dps soooo much that we can't leave them. But the problem is you'll have mana to burn more frequently. And it's bad. We're already mana hungry when we use our skills, so we don't want it. Pick some Greater Glyph of Mana Regeneration for counter this.

Greater Quintessence of Movement Speed ... what was you expecting ? Didn't you read to title ?!?

Marks and Quint won't move, because they give us too much for changing them.

We're just going to change the flat armor by scaling armor for seals.

We want some magic resistance, so we'll pick some Greater Glyph of Magic Resist. In that case, less cooldown reduction means less mana regeneration needed. I'll let you choose your own mix :)

greater mark of hybrid penetration : You can use it if you don't like the 8.5ad and you prefer some late game efficiency.
Greater Seal of Health You can use it against an Ap top for being more tanky early, but I like to take less damage from ganks/minions/AA
Greater Quintessence of Health If you feel like the little bonus movespeed isn't helping you, of if you have some issue for staying alive, pick those
Greater Quintessence of Experience If you like them, go with it ! If you can harass your opponent pre-6, wait to be lvl 6 and him to stay at lvl 5 and just do your combo.

Their is not that much items you can pick at lvl one. I think Cloth Armor and 5x Health Potion or Crystalline Flask and 3x Health Potion are the only thing you can pick. They are both a great choice, I'm not just taking the Flask because the armour can't help us.
- With the Cloth Armor you'll be able to stand longer in lane with trading against some Zed, Tryndamere, Fiora etc... and the potions prevent poke/lack in trade pre-sheen.
- But, with the Crystalline Flask you have a bit less sustain (yeah, you win 60hp but you loose 15armor...) but you win some nice 180mana ! With this mana, you'll be able to reach some Kayle, Vladimir or Elise who are out of our Q. Just E-Q-wall back, you're 180mana will sustain it.

Generally, you want to take boots on purpose of your laning opponent so Ninja Tabi and Mercury's Treads will often by yours boots. But if you are dominating like a real smash-smurf you can also pick Boots of Swiftness. In late game, if they don't have so much CC buy the Boots of Swiftness and if you don't have to defend the base, pick Enchantment: Alacrity if you have to, pick Enchantment: Homeguard

At the very first place, we NEED that Sheen/ Trinity Force proc. The Iceborn Gauntlet does a little bit less damage than Trinity, and helps way more to chase BUT as poppy, we benefit so hard from ALL the stats or the Trinity. So leave this Gauntlet here.

Then, as Poppy, we need sustain. The three item poppy could use here are : the old nice Bloodthirster, the Hextech Gunblade and the Blade of the Ruined King. Let's take a look at them

The blood is really nice and we can protect theses stack pretty easily with our survivability but it didn't got any CC :/

This one is really nice because it give you the best sustain of the three with this spell vamp and a limited CC with the active, but this is the most expensive

I take this one because it give us a massive CC with a medium life steal, but this item will does a huge amount of damage on tanks, and on baron/dragon.

We want to reach the carry to most quickly possible, but feel free to change if you don't like the blade (but try it first :) ).

Here we can use well the 5% boost from the Zeal but upgrading it won't give us more speed, and at this point, Poppy want to upgrade her Paragon of Demacia (shield). And remember, you can hurt someone badly with a shield. So I'm listing all the shields and armour you can use :

- Warmog's Armor This one is really nice, it give you 1k of hp ! It make you tanky, but same as Guardian Angel people doesn't tend to focus someone who have the revive active or lot of hp. But remember that Armor and Magic Resist works better with your passive.

- Guardian Angel. As I said it above, it give you a really nice mental advantage because they won't tend to focus someone will revive after that (even more if this person can ult someone can kill the others) and it give you some nice resist over all :) Don't take it if you want the enemy to focus you.

- Randuin's Omen This is the perfect item to counter team pretty equilibrate with a huge ad top like Riven, Zed, Fiora or Renekton, because you can build a early Warden's Mail. (Don't tell him I told you this, but it's the perfect item if they have a fed Darius, because it counters his ult and his damage)

- Frozen Heart Yay ! It give us some Cooldown reduction and some mana in bonus ! This item sounds kinda like Thornmail : give you more bonus but doesn't kill Tryndamere, Master Yi ... So take it if they got lots of AD that doesn't rely that much on AA.

- Thornmail It give you no bonus other than armor but it destroy anyone who relies one AA. As I said it upper, take it when you have Master Yi, Tryndamere, Fiora ? etc...

- Runic Bulwark This is the best item against a huge AP team, but generally someone will take it too. Coordinate yourself with your team to don't pick two of theses. If someone want to take it too, pick up the Maw of Malmortius instead :).

- Maw of Malmortius ... Look up ! Pick it if you want more damage or if someone took the bulwark

Now we have our defensive item, we want more damage ! After taking our damage-item we will, of course, end our zeal :)
I'm personally picking the Infinity Edge because we already have 20% Crit chance, and we'll have 30% with our Phantom Dancer. I don't think I need much sustain, but for being honest, I almost never end my build with AD Speed Poppy... But If you find a game where you will go that far, and if you don't like the IE, there is some alternatives choices :

- Blade of the Ruined King / Hextech Gunblade / Bloodthirster If you want more sustain, you can pick it (don't pick twice the same). If I had to choose, I would take the the blade then the hextech gunblade.

- Last Whisper / Black Cleaver I said in the Runes - Marks that our damage output
doesn't come only from Ad, so picking one of them won't help that much. If you still take that, remember : Black Cleaver is better when they have less than ~110 armor, and if higher take the Last Whisper.

- Ravenous Hydra Poppy isn't brilliant at farming, and can't really split push. But with this you can split push easily and get away pretty much with no problem. But I don't think you might want to push with Poppy (If you can deal an AoE Q, take it !!!! I'll try it out)

- Mercurial Scimitar If they have a team with a lot of CC you could buy it, if they have some AP fed too, it's even better. And at least you can use this short boost of speed to reach them faster.

- Youmuu's Ghostblade The active is very nice, and you benefit from all the stats of this, but I always forgot to use it, so I don't take it ^^".