As Tinstaafl says, certain combinations of height and airspeed result in too little enegry in the system as a whole ( combined kinetic and potential ) to make a safe flare and have enough left for the cushion at the bottom of an autorotation.
Single engined take off profiles follow a narrow corridor outside the high/slow and low/fast portions of the avoid curve. In an R22, this is basically an acceleration at 5' to 10' agl, rotate at 40kts, then climb away at 60kts.
Towering ( vertical ) takeoffs are available in certain circumstances - especially if you're in such a confined area that you've no room to accelerate. However, an engine failure will mean some seriously quick reactions and even that won't guarantee a successful outcome.
As for the best ROC speed, imagine an upside down bell-shaped curve where the faster you go, up to a certain speed, you need less power for straight and level. Above that point you need to add power to stay straight and level. 53kts in an R22 is the bottom of the curve, and therefore where most power is left for climbing.
Ask on the Rotorheads forum if you want more - there's far more knowledgeable people than me there.
[Edited for spellin]
[ 12 July 2001: Message edited by: The Nr Fairy ]