The three builds compared are:
(Current) Blade of the Ruined King + Runaan's Hurricane vs.
Bloodthirster + Infinity Edge vs.
Phantom Dancer + Infinity Edge

Assumptions:
Level 15 Tristana with rank 4 Rapid Fire
Berzerker's Greaves
0 runes/masteries
Pre-mitigation damage (targets have 0 armor)

The next two assumptions are for averaging reasons:
BT build will have 20 of the possible 30 stacks (so +90 AD)
BotRK build will be hitting 1 primary and 1 secondary target each with a constant 1100 HP

Costs:
BotRK + Hurricane = 5650g
BT + IE = 7000g
PD + IE = 6600g

Average Damage per shot:
Tristana with no Berzerker's: 98 DMG
BotRK/Hurricane: 188.5 primary + 131.75 secondary = 320 DMG
BT/IE: 248.5 * (crit chance & multiplier: 1.375) = 342 DMG
PD/IE: 158.5 * (crit chance & multiplier: 1.825) = 289 DMG

Average Damage per second:
Tristana with no Berzerker's: 91 DPS
BotRK/Hurricane: 288 primary + 201 secondary = 489 DPS
BT/IE: 365 DPS
PD/IE: 404 DPS

Average Damage per second with Rapid Fire 4:
Tristana with no Berzerker's: 139 DPS
BotRK/Hurricane: 381 primary + 266 secondary = 647 DPS
BT/IE: 533 DPS
PD/IE: 547 DPS

I have more detailed math behind these results if anyone has anything specific they want to see.

Wouldn't BotRK deal less damage the less HP the target(s) have/has?

If you're constantly in range to hit 3 targets at once, there'e probably a fed irelia with several blades up your hind end.

So what I'm getting from this is BotRK/Hurricane has substantially less single target DPS than the other two. You're only putting 380dps on one target and 250 on a 2nd target. So you could kill the first target almost twice as fast with PD/IE and just move onto the next target, imo. Which is actually closer to what team fights usually are anyway, in my experience.

Your job as an ADC is to take out the front lines, 1 by 1. Not do AOE damage, AOE damage is to be fulfilled by your APC/AD casters.

Any good ADC will tell you its best to do the most possible damage to one target than any other.

That said, thanks for the math on getting IE-> PD vs IE + BT I've always wondered the math, though I've always rushed BT/IE on Trist because of her AS Steroid. IE is usually after LW/Warmog anymore.

I would think hurricane isn't the best on tristana, since the only on-hit effects are her e passive, unless the BoRK also counts as on-hit.

This just in: hitting multiple targets at the same speed as hitting one target does more DPS!

Results: BotRK and Runnan's are viable!

Flawless logic!